Imaging polarimetry

ABSTRACT

To effectively reduce a measurement error in a parameter indicating two-dimensional spatial distribution of a state of polarization generated by variations in retardation of a birefringent prism pair due to a temperature change or other factors, while holding a variety of properties of an imaging polarimetry using the birefringent prism pair. By noting that reference phase functions φ 1 (x, y) and φ 2 (x, y) are obtained by solving an equation from each vibration component contained in an intensity distribution I(x, y), the reference phase functions φ 1 (x, y) and φ 2 (x, y) are calibrated concurrently with measurement of two-dimensional spatial distribution S 0 (x, y), S 1 (x, y), S 2 (x, y), and S 3 (x, y) of Stokes parameters.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an imaging polarimetry in which, and animaging polarimeter with which, a two-dimensional spatial distributionof a state of polarization of light under measurement is measured by theuse of a birefringent prism pair.

2. Description of the Related Art

Light has properties of a “transverse wave”. Based upon the premise ofthree mutually orthogonal axes (x, y, z), when a propagation directionof light is assumed to be the z-axis direction, a vibration direction ofthe light is a direction along the x-y plane. The vibration direction ofthe light within the x-y plane has a bias. This bias of light isreferred to as “polarization”. A biased state of light is referred to asa “state of polarization (SOP)” in this specification. Typically, theSOP varies depending upon positions (coordinates) in the two-dimensionalx-y plane.

When light in some state of polarization is incident on an object undermeasurement to acquire emitted light such as transparent or reflectedlight and the object under measurement has optical anisotropy to thelight, a change in SOP is observed between incident light and emittinglight. Acquiring information on anisotropy of the object undermeasurement from the change in SOP is referred to as “polarimetry”. Itis to be noted that causes of such anisotropy may include anisotropy ofa molecular structure, presence of stress (pressure), and presence of alocal field and a magnetic field.

A measurement in which a change in SOP between the incident light andthe emitted light is obtained with respect to each position(coordinates) of the two-dimensional x-y plane and then to acquireinformation on anisotropy of the object under measurement is especiallyreferred to as “imaging polarimetry”. This imaging polarimetry has anadvantage of acquiring a great amount of information as compared to thecase of measurement at a point or in a region averaged by a face in thex-y plane. In the imaging polarimetry, a device for measuring an SOP ofemitted light (occasionally, incident light), namely an imagingpolarimeter, is a key device.

As fields of application of the imaging polarimetry known are the fieldof an inspection of optical electronics, the medical field, the remotesensing field, the machine vision field and the like. In the field ofthe inspection of the optical electronics, for example, sincebirefringence or a defect due to residual stress can be measured in anondestructive and non-contact manner, the imaging polarimetry has beenapplied to an inspection or study of a liquid crystal, an optical film,an optical disc, and the like. In the medical field, an attempt has beenmade for early detection of glaucoma or a cancer cell since severalkinds of cells have polarization properties. In the remote sensingfield, an inclination or the degree of flatness of an object undermeasurement can be measured from the two-dimensional spatialdistribution of the state of polarization by remote control and forexample, the imaging polarimetry is applied to an examination invegetation. In addition, in the machine vision field, for the samereason, a configuration of an object is recognized from a polarizationimage.

Incidentally, assuming that light traveling in the z-axis directionexists, polarized light in a state where a vibration component in thex-axis direction is perfectly correlated (synchronized) with a vibrationcomponent in the y-axis direction is classified into three types:linearly polarized light, elliptically polarized light, and circularlypolarized light. Parameters for expressing a state of ellipticallypolarized light are: ε for an ellipticity angle, θ for an azimuth angle,Δ for a phase difference, and Ψ for an amplitude ratio angle.

Further, as parameters for effectively expressing a degree ofpolarization of light, the ellipticity angle, the azimuth angle and thelike, Stokes Parameters are used. The Stokes Parameters are composed offour parameters having definitions as follows:

S₀: total intensity

S₁: difference between intensities of linearly polarized components withangles of 0° and 90°.

S₂: difference between intensities of linear polarized components withangles ±45°.

S₃: difference between intensities of left and right circularlypolarized light components.

In a three-dimensional space where the three mutually orthogonal axesare taken as S₁, S₂ and S₃, assuming a sphere with a radius S₀ and anoriginal point of the axes taken as a center, an SOP of arbitrary lightis expressed as one point in this three-dimensional space and a degreeof polarization is expressed by the following expression:$\begin{matrix}{{{Degree}\quad{of}\quad{polarization}} = \left( {{distance}\quad{from}\quad{original}\quad{point}\quad{to}\quad{point}}\quad \right.} \\{\left( {S_{1},S_{2},S_{3}} \right)/S_{0}} \\{= {\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{1/2}/S_{0}}}\end{matrix}$

It may be understood from the above that in the case of a perfectlypolarized light (degree of polarization=1), one point expressing the SOPexists in the sphere with a radius S₀. Further, the ellipticity angleand the azimuth angle respectively correspond to halves of a latitudeand a longitude of the one point expressing the SOP in the abovethree-dimensional space. As thus described, it is possible to expressall information on the SOP if the four parameters S₁, S₂, S₃ and S₀ ofthe Stokes Parameters can be obtained.

As conventionally prevailing imaging polarimetries, a rotating-retarderpolarimetry and a polarization-modulation polarimetry are known.

In the rotating-retarder polarimetry, a retarder and an analyzerintervene in sequence in a channel for light under measurement toward animaging device. Here, the retarder is an optical element having twoprincipal axes (fast axis and slow axis) in mutually orthogonaldirections, and is also configured to change a phase difference betweenthe two principal axes before and after passage of light. Further, theanalyzer is an optical element having one principal axis and also isconfigured to allow transmission of only one linearly polarized lightcomponent corresponding to the direction of the principal axis.

In this rotating-retarder polarimetry, for obtaining two-dimensionalspatial distributions of the four Stokes Parameters independently, it isnecessary to physically rotate a retarder itself and take an intensitydistribution measurement for at least four kinds of directions. Namely,the Stokes Parameters of incident light are expressed as functions S₀(x,y), S₁(x, y), S₂(x, y), and S₃(x, y) of two-dimensional spatialcoordinates.

In the polarization-modulation polarimetry, two retarders (firstretarder and second retarder) capable of electrically controlling aphase difference and one analyzer intervene in sequence in a channel forlight under measurement toward an imaging device. Among such retardersused are an electro-optic modulator, a liquid crystal and a photoelasticmodulator. For example, a phase difference of 45° is set between theprincipal axes of the first retarder and the second retarder.

Also in this polarization-modulation polarimetry, for obtainingtwo-dimensional spatial distributions of the four Stokes Parametersindependently, it is necessary to vibrate, by electric control, a phasedifference between the first retarder and the second retarder in apredetermined angle range to obtain a plurality of intensitydistributions.

However, concerning the conventional general imaging polarimetrytypified by the rotating-retarder polarimetry and thepolarization-modulation polarimetry, the following problems have beenpointed out.

(1) First Problem

Since a mechanical or active polarization controlling element isrequired, there are problems including that: [1] a problem of vibration,heat generation and the like are unavoidable; [2] the degree of sizereduction is limited due to necessity for a mechanical element and thelike to have some capacity; [3] a driving device for consuming electricpower is essential; and [4] maintenance is necessary and complex.

(2) Second Problem

Since it is necessary to repeatedly measure a plurality of intensitydistributions while changing conditions of the polarization modulating(controlling) element, there are problems including that: [1]measurement takes relatively long; and [2] an object under measurementneeds to be kept stable during measurement.

In order to solve the above problems with the conventional generalimaging polarimetry, the present inventors and the like developed, inadvance, an “imaging polarimetry using a birefringent prism pair” (referto T. Kaneko and K. Oka, “Measurement of spatial two-dimensionaldistribution of polarized light state using birefringent wedge,” The49th Extended Abstracts, Japan Society of Applied Physics and RelatedSocieties (Japan Society of Applied Physics, Hiratsuka, 2002) p. 977 andK. Oka and T. Kaneko, “Compact complete imaging polarimeter usingbirefringent wedge prisms,” Opt. Express, Vol. 11, No. 13, pp.1510-1519, 2003).

A constitutional view of an experiment system for explaining the imagingpolarimetry using the birefringent prism pair is shown in FIG. 19. Asapparent from this figure, light projected from a helium-neon laser 1 isenlarged in its beam diameter by collimator lenses 2 and 4 and a pinhole3 and transmitted through a polarizer 5 and a twisted nematic liquidcrystal 6, to obtain a light wave having an SOP depending upon theposition (coordinates) of the two-dimensional x-y plane. Two-dimensionalspatial distributions S₀(x, y), S₁(x, y), S₂(x, y) and S₃(x, y) of theStokes parameters of the light wave are obtained by a measurement system7 surrounded with a broken line in the figure.

Light under measurement is first transmitted through an imaging lens 8,and through two birefringent prism pairs BPP₁ and BPP₂ and a flat-plateanalyzer A in sequence, and then incident on a CCD imaging element 9.The image lens is used in order to forms an image of a projectionsurface of the twisted nematic liquid crystal 6 on the CCD imagingelement. Meanwhile, the two birefringent prism pairs BPP₁ and BPP₂, andthe flat-plate analyzer A are overlapped on a front face of the CCDimaging element. (The surfaces of BPP₁ and BPP₂ and A may be focused onthe front surface of the CCD imaging element optically in the use of arelay lens and the like. The birefringent prism pair comprises a pair ofwedge-shaped prisms formed of a birefringent medium and alternatelyoverlapped with each other. A contact surface of each of the twobirefringent prism pairs BPP₁ and BPP₂ is inclined at a fine angle withrespect to x axis and y axis. Here, two principal axes of thebirefringent prism pair BPP₁ agree with the x and y axes, while twoprincipal axes of the BPP₂ are inclined 45° from those. Here, atransmission axis of the analyzer A is arranged in parallel to the xaxis.

In each of the two birefringent prism pairs BPP₁ and BPP₂, a phasedifference created between the orthogonal polarized light componentsdepends upon two-dimensional spatial coordinates. Hence, as shown inFIG. 20, an intensity distribution including three carrier components isobtained from the CCD imaging element 9. An amplitude and a phase ofeach of the carrier components are modulated by the two-dimensionalspatial distribution of the Stokes Parameters of the light undermeasurement. It is therefore possible to obtain each of the StokesParameters by execution of a signal processing with a computer 10 by theuse of Fourier transformation.

One example of results of an experiment is shown in FIG. 21. This is aresult obtained in the case of uniformly applying an electric field toonly a part of a transparent electrode of a character “A” in the twistednematic liquid crystal 6. Both right and left figures showtwo-dimensional spatial distribution θ (x, y) of the azimuth angle andtwo-dimensional spatial distribution ε (x, y) of the ellipticity anglewhich are calculated from the two-dimensional spatial distribution ofthe Stokes parameters, respectively. It is thereby understood that anSOP depends upon two-dimensional spatial coordinates.

As thus described, according to the imaging polarimetry using thebirefringent prism pair, it is possible to obtain the two-dimensionalspatial distribution of each of the Stokes Parameters by a frequencyanalysis of properties of the intensity distribution. It is reasonablynecessary to obtain respective retardations of the two birefringentprism pairs BPP₁ and BPP₂ prior to the frequency analysis. Here,retardation means a phase difference created between linearly polarizedlight components along the two orthogonal principal axes.

According to the foregoing imaging polarimetry using the birefringentprism pair, advantages can be obtained including that: [1] amechanically movable element such as a rotating retarder is unnecessary;[2] an active element such as an electro-optic modulator is unnecessary;[3] four Stokes Parameters can be obtained from one intensitydistribution at once so that a so-called snap shot measurement can beperformed; and [4] the constitution is simple, and thus suitable forsize reduction.

However, concerning the foregoing imaging polarimetry using thebirefringent prism pair, a problem of generation of a relatively largemeasurement error has been pointed out for the following reasons.

(1) Variations (Fluctuations) in Retardation of the Birefringent PrismPairs BPP₁, BPP₂

Retardation of the birefringent prism pair varies sensitively due to atemperature or pressure change, resulting in that the phase of theintensity distribution detected by the imaging element varies due to thetemperature or pressure change, as shown in FIG. 22. Consequently, asshown in FIG. 23, the temperature or pressure change causes generationof an error in a measured value of the Stokes parameter obtained fromthe intensity distribution. In addition, although only the x surface isshown in FIGS. 22 and 23 for the sake of simplicity, the same is said inthe y direction.

(2) Displacement in Relative Position Between the Birefringent PrismPair and the Imaging Element

In a system to which a relay lens is inserted between the birefringentprism pair and the imaging element, relative positional displacementbetween both of them causes a large error factor. When the coordinateson the birefringent prism pair to be sampled by each pixel of theimaging element is displaced due to the vibration every measurement, asshown in FIG. 24, a state is generated which is equivalent to a casewhere retardation of the birefringent prism pair varies, resulting ingeneration of an error in a measured value of the Stokes parametersobtained from the intensity distribution. In addition, although only thex surface is shown in FIG. 24 for the sake of simplicity, the same issaid in the y direction.

Incidentally, for example, in the inspection of the optical electronics,accuracy required in a two-dimensional spatial distribution of anellipticity angle or an azimuth angle is considered to be an error inthe order of not larger than 0.1°. When this accuracy is to be realizedby stabilizing retardation of the birefringent prism pair, it isnecessary to keep a variation in temperature of the birefringent prismpair at or under 0.5° C.

However, it requires a large-sized temperature compensating device suchas a heater or a cooler for the temperature stabilization, whichunfavorably causes a loss of advantages (size reduction, non-inclusionof an active element, etc.) of the imaging polarimetry using thebirefringent prism pair. Hence it is practically difficult to reduce ameasurement error by stabilizing the retardation of the birefringentprism pair.

In addition, in the system to which the relay lens is inserted, it ispractically difficult to prevent the vibration so that the relativedisplacement between the birefringent prism pair and the imaging elementbecome negligible in the applied field in which the polarimeter has tobe provided on a mobile body such as the remote sensing or the robotvision field.

SUMMARY OF THE INVENTION

The present invention was made by noting the problems of theconventional imaging polarimetry using a birefringent prism pair. It isan object of the present invention to provide an imaging polarimetry andan imaging polarimeter using a birefringent prism pair, which arecapable of measurement with higher accuracy, while holding theadvantages thereof including that: a mechanically movable element suchas a rotating retarder is unnecessary; an active element such as anelectro-optic modulator is unnecessary; four Stokes Parameters can beprovided from one intensity distribution at once so that a so-calledsnap shot measurement can be performed; and the constitution is simple,and thus suitable for size reduction.

Further objects and working effects of the present invention are readilyunderstood by the skilled in the art by referring to the followingdescription of the specification.

(1) An imaging polarimetry of the present invention comprises: a step ofpreparing a polarimetric imaging device, a step of obtaining atwo-dimensional intensity distribution, and an arithmetic step.

A polarimetric imaging device provided in the step of preparing apolarimetric imaging device is one where a first birefringent prismpair, a second birefringent prism pair and an analyzer, through whichlight under measurement passes in sequence, and a device for obtaining atwo-dimensional intensity distribution of the light having passedthrough the analyzer are provided, each birefringent prism paircomprises parallel flat plates in which two wedge-shaped retardershaving the same apex angle are attached and directions of fast axes ofthe two retarders are orthogonal to each other, the second birefringentprism pair is arranged such that the direction of a principal axis ofthe second birefringent prism pair disagrees with the direction of aprincipal axis of the first birefringent prism pair, and the analyzer isarranged such that the direction of a transmission axis of the analyzerdisagrees with the direction of the principal axis of the secondbirefringent prism pair.

In the step of obtaining the two-dimensional intensity distribution, thelight under measurement is launched into the polarimetric imagingdevice.

In the arithmetic step, by the use of the obtained two-dimensionalintensity distribution, a set of phase attribute functions of ameasurement system is obtained, and also a parameter indicating atwo-dimensional spatial distribution of a state of polarization (SOP) ofthe light under measurement is obtained. Here, the set of phaseattribute functions is a set of functions defined by properties of thepolarimetric imaging device, and includes a function depending upon atleast a first reference phase function (φ₁(x, y)) as retardation of thefirst birefringent prism pair and a function depending upon at least asecond reference phase function (φ₂(x, y)) as retardation of the secondbirefringent prism pair, and by those functions themselves, or byaddition of another function defined by the properties of thepolarimetric imaging device, the set of phase attribute functionsbecomes a set of functions sufficient to determine the parameterindicating the two-dimensional spatial distribution of the SOP of thelight under measurement.

There may be a case where a one-dimensional or two-dimensional imagingelement is used as the “means of obtaining a two-dimensional intensitydistribution”. The one-dimensional imaging element can be used when thespatial distribution of the SOP of the light under measurement and thepolarimetric imaging device are relatively displaced such as a casewhere the object under measurement and the polarimetric imaging deviceare relatively displaced almost perpendicular to the propagationdirection of the light under measurement, for example. When suchrelative displacement is used, the two-dimensional intensitydistribution can be obtained from a change in time of the intensityobtained from the one-dimensional imaging element. At this time, it isnecessary that the arrangement direction of the pixels of theone-dimensional imaging element has to disagree with the direction ofthe above relative displacement. For example, that arrangement directionmay be perpendicular to the direction of the above relativedisplacement.

“Obtaining a parameter indicating a two-dimensional spatial distributionof an SOP” includes obtaining all or part of two-dimensional spatialdistributions of the four Stokes parameters, namely, S₀(x, y) forexpressing a total intensity, S₁(x, y) for expressing a differencebetween intensities of linear polarized light components with angles of0° and 90°, S₂(x, y) for expressing a difference between intensities oflinear polarized light components with angles ±45, and S₃(x, y) forexpressing a difference between intensities of the left-hand andright-hand circularly polarized light components. While whether all thetwo-dimensional spatial distributions of the Stokes parameters areobtained or not is left to a person executing this step, all thetwo-dimensional spatial distributions of the Stokes parameters can beobtained in principle according to the present invention.

Further, “obtaining a parameter indicating a two-dimensional spatialdistribution of an SOP” includes the case of obtaining a parameterequivalent to the two-dimensional spatial distribution of the Stokesparameter. For example, a two-dimensional spatial distribution of a setof parameters of a light intensity, a degree of polarization, anellipticity angle and an azimuth angle, or a two-dimensional spatialdistribution of a set of parameters of a light intensity, a degree ofpolarization, a phase difference, and an amplitude ratio angle, isequivalent to the two-dimensional spatial distribution of the Stokesparameters. While all of these parameters can be obtained in principleaccording to the present invention, the above-mentioned obtainment of aparameter also includes a case where part of the parameters is obtainedby selection of a person executing the step.

“Another function defined by the properties of the polarimetric imagingdevice”, can be corresponded to a reference amplitude function, areference value for calibration of a reference phase function, datashowing a relation between the first reference phase function and thesecond reference phase function, data showing a relation between thefirst reference phase function difference and the second reference phasefunction difference, and the like.

When the two-dimensional intensity distribution of the SOP of the lightin a specific region under measurement such as a region occupied by theobject under measurement is to be measured, the two-dimensional spatialdistribution of the SOP of the light in that region is to be reproducedby the polarimetric imaging device. As one means for that, an imaginglens is provided between the region under measurement and thepolarimetric imaging device so that the region under measurement isfocused on the polarimetric imaging device. At this time, it ispreferable that the first birefringent prism pair, the secondbirefringent prism pair, the analyzer, and the imaging element areprovided in proximity to each other and these are within a focus depthof the imaging lens. Alternatively, the second birefringent prism pairand the imaging element may be set apart from each other and a relaylens may be provided so that the image of the region under measurementwhich was focused on the first birefringent prism pair and the secondbirefringent prism pair by the imaging lens can be focused on theimaging element by the relay lens again. In this case, the analyzer maybe provided at any place between the second birefringent prism pair andthe imaging element.

According to another means for reproducing the two-dimensional spatialdistribution of the polarization properties in the region undermeasurement, in the polarimetric imaging device, the region undermeasurement and the imaging device are provided close to each other andthe region under measurement is projected on to the imaging device byparallel light beams without providing the imaging lens. Namely, toprovide the imaging lens is not essential in the present invention.

The present invention is not limited to measurement of thetwo-dimensional spatial distribution of the SOP of the light in theregion under measurement, and it can be applied to measurement ofpropagation direction distribution of SOP of light under measurementwith an optical system which converts a distribution of a propagationdirection of light under measurement to a two-dimensional spatialdistribution of light in an imaging device, for example. Suchmeasurement can be implemented by providing a lens between the regionunder measurement and the imaging device and setting the region undermeasurement so as to be focused by the lens.

According to the image polarimetry of the present invention, ameasurement error in the parameter indicating the two-dimensionalspatial distribution of the SOP, generated by variations in retardationof the birefringent prism pair due to a temperature change or otherfactors can be effectively reduced while properties of the imagingpolarimetry using the birefringent prism pair is remained in which theparameters showing the two-dimensional spatial distribution of the allSOP of the light under measurement can be obtained in principle byobtaining the intensity distribution once without needing a mechanicallymovable part for controlling polarized light and an active element suchas an electro-optical modulator.

(2) The analyzer may be arranged such that the direction of thetransmission axis thereof forms an angle of 45° with respect to thedirection of the principal axis of the second birefringent prism pair.

(3) In one embodiment of the imaging polarimetry of the presentinvention, the set of the phase attribute functions is composed of thefirst reference phase function and the second reference phase function.In the arithmetic step of this embodiment, data showing a relationbetween the first reference phase function and the second referencephase function is made available.

The arithmetic step according to this embodiment is a unit where, by theuse of the obtained intensity distribution, a first intensitydistribution component which nonperiodically vibrates with specialcoordinates and a third intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon the secondreference phase function and not depending upon the first referencephase function are obtained, and at least one of a second intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon a difference between the first reference phasefunction and the second reference phase function, a fourth intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon a sum of the first reference phase function andthe second reference phase function, and a fifth intensity distributioncomponent which vibrates with spatial coordinates at a frequencydepending upon the first reference phase function and not depending uponthe second reference phase function is obtained, and by the use of thedata showing the relation between the first reference phase function andthe second reference phase function and each of the intensitydistribution components, the first reference phase function and thesecond reference phase function are obtained, and also the parameterindicating the two-dimensional spatial distribution of the SOP isobtained.

Here, the “data showing the relation between the first reference phasefunction and the second reference phase function” is data with which oneof the two reference phase functions can be obtained when the otherthereof is given, such as ratios of inclination between the tworeference phase functions with respect to x axis and y axis.

“Obtaining the reference phase function” includes the case of obtaininga parameter equivalent thereto. In particular, obtaining a complexfunction including information on the reference phase functioncorresponds to obtaining a parameter equivalent to the reference phasefunction.

In a case where the direction of the transmission axis of the analyzerforms an angle of 45° with respect to the direction of the principalaxis of the second birefringent prism pair, the fifth spectral intensitycomponent does not appear. Therefore, when at least one of the second,fourth and fifth intensity distribution components is to be obtained inthe arithmetic step, at least either the second intensity distributioncomponent or the fourth intensity distribution component may beobtained. In this manner, there is an advantage of making the arithmeticoperation simpler. Meanwhile, in the case of not limiting the anglebetween the direction of the transmission axis of the analyzer and thedirection of the principal axis of the second birefringent prism pair toan angle of 45°, there is an advantage of easing a limitation on anerror in assembly of an optical system, to facilitate manufacturing ofthe optical system.

(4) In another embodiment of the imaging spectrometry of the presentinvention, the set of phase attribute functions is composed of adifference (Δφ₁(x, y)) of the first reference phase function from areference value for calibration of the first reference phase functionand a difference (Δφ₂(x, y)) of the second reference phase function froma reference value for calibration of the second reference phasefunction. In the arithmetic step of this embodiment, the reference value(φ₁ ^((i))(x, y)) for calibration of the first reference phase function,the reference value (φ₂ ^((i))(x, y)) for calibration of the secondreference phase function, and data showing a relation between the firstreference phase function difference and the second reference phasefunction difference are made available.

The arithmetic step according to this embodiment is a unit where, by theuse of the obtained intensity distribution, a first spectral intensitycomponent which nonperiodically vibrates with spatial coordinates and athird intensity distribution component which vibrates with spatialcoordinates at a frequency depending upon the second reference phasefunction and not depending upon the first reference phase function areobtained, and at least one of a second intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon adifference between the first reference phase function and the secondreference phase function, a fourth intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon asum of the first reference phase function and the second reference phasefunction, and a fifth intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon the firstreference phase function and not depending upon the second referencephase function is obtained, and by the use of the reference value forcalibration of the first reference phase function, the reference valuefor calibration of the second reference phase function, the data showingthe relation between the first reference phase function difference andthe second reference phase function difference, and each of the obtainedintensity distribution components, the first reference phase functiondifference and the second reference phase function difference areobtained, and also the parameter indicating the two-dimensional spatialdistribution of the SOP is obtained.

Here, the “reference values for calibration” of the first and secondreference phase functions may be measured initial values of therespective reference phase functions, or may be appropriately set valuesthereof not based upon actual measurement. However, the relation betweenthe two reference values for calibration preferably conforms to therelation between the first reference phase function difference and thesecond reference phase function difference.

The “reference phase function difference” is defined as a differencebetween the “reference phase function” and the “reference value forcalibration of the reference phase function”. When the “reference valuefor calibration of the reference phase function” does not agree with anactual initial value, therefore, the “reference phase functiondifference” does not mean an actual difference in the reference phasefunction.

The “data showing the relation between the first reference phasefunction difference and the second reference phase function difference”is data with which one of the two reference phase function differencescan be obtained when the other thereof is given, such as ratios ofinclination between the two reference phase function differences withrespect to x axis and y axis.

“Obtaining the reference phase function difference” includes the case ofobtaining a parameter equivalent thereto. In particular, obtaining acomplex function including information on the reference phase functiondifference corresponds to obtaining a parameter equivalent to thereference phase function difference.

As for the case where the fifth intensity distribution component doesnot appear when the direction of the transmission axis of the analyzerforms an angle of 45° with respect to the direction of the principalaxis of the second birefringent prism pair, the foregoing case applies.

For comparison with this embodiment, first, the case of proceeding anarithmetic operation not by the use of the reference phase functiondifference but by the use of the reference phase function as in theembodiment of (3) above is considered. Putting aside appearance in anarithmetic operation, the second reference phase function, which is inprinciple determinable independently from an SOP of the light undermeasurement, is first determined, and subsequently the first referencephase function is determined using the second reference phase function.At this time, the obtained second reference phase function isaccompanied with phase ambiguity of an integral multiple of 2π. Suchaccompanying phase ambiguity itself does not affect a calculation errorin parameters showing the two-dimensional spatial distribution of theSOP. However, since phase unwrapping performed in obtaining the firstreference phase function from the second reference phase function causesgeneration of a calculation error in the first reference phase function,a calculation error in the parameters showing the two-dimensionalspatial distribution of the SOP may be generated. Phase unwrapping is aprocess of determining a value of the second reference phase functionsuch that the value of the second reference phase function continuouslychanges beyond the range of 2π with respect to a position change. In thecase of not using the second reference phase function difference, thefirst reference phase function is obtained through the use of the “datashowing the relation between the first and second reference phasefunctions” to the second reference phase function after phaseunwrapping. When the position intervals at the time of change in thevalue of the second reference phase function by 2π are not sufficientlylarge as compared to sampling intervals of the position, or when a noiseis included in the measured value of the second reference phasefunction, the second reference phase function after phase unwrappingcould be calculated by a wrong unit, 2π. If the first reference phasefunction is obtained from the second reference phase function includingthe error by the unit of 2π, since an error included in the firstreference phase function is typically not calculated by the unit of 2π,the error in the first reference phase function would become a largeerror in the case of calculating the parameters indicating thetwo-dimensional spatial distribution of the SOP. As opposed to this, inthe case of the embodiment of (4) above, since the second referencephase function difference changes modestly with respect to the positionchange, phase unwrapping on the second reference phase functiondifference is unnecessary or necessary only in a small frequency,thereby leading to elimination of, or extreme reduction in, thepossibility for generation of an error in the first reference phasefunction difference due to phase unwrapping.

(5) In the embodiment of (4) above, the spectroscopic polarimetry mayfurther comprise a step of launching light for calibration, with knownparameters each showing the two-dimensional spatial distribution of theSOP, into the polarimetric imaging device to obtain a two-dimensionalintensity distribution for calibration, so as to obtain the referencevalue (φ₃ ^((i))(x, y)) for calibration of the first reference phasefunction and the reference value (φ₂ ^((i))(x, y)) for calibration forthe second reference phase function by the use of each of the parametersshowing the two-dimensional spatial distribution of the SOP of the lightfor calibration and the obtained intensity distribution for calibration,whereby these reference values for calibration are made available.

(6) Moreover, in the embodiment of (4) above, the imaging polarimetrymay further comprise a step of launching light for calibration, withknown parameters each showing the two-dimensional spatial distributionof the SOP, into the polarimetric imaging device to obtain atwo-dimensional intensity distribution for calibration, so as to obtainthe reference value (φ₁ ^((i))(x, y)) for calibration of the firstreference phase function, the reference value (φ₂ ^((i))(x, y)) forcalibration for the second reference phase function, and the datashowing the relation between the first reference phase functiondifference and the second reference phase function difference, by theuse of each of the parameters showing the two-dimensional spatialdistribution of the SOP of the light for calibration and the obtainedintensity distribution for calibration, whereby these reference valuesfor calibration are made available.

(7) In the embodiment of (3) above, the imaging polarimetry may furthercomprise a step of launching light for calibration, with knownparameters each showing the two-dimensional spatial distribution of theSOP, into the polarimetric imaging device to obtain a two-dimensionalintensity distribution for calibration, so as to obtain the data showingthe relation between the first reference phase function difference andthe second reference phase function difference, by the use of each ofthe parameters showing the two-dimensional spatial distribution of theSOP of the light for calibration and the obtained intensity distributionfor calibration, whereby the data showing the relation between the firstreference phase function difference and the second reference phasefunction difference is made available.

(8) In the embodiments of (5) and (6) above, it is possible to uselinearly polarized light as the light for calibration.

(9) In the embodiment of (7) above, it is possible to use linearlypolarized light as the light for calibration.

(10) In another embodiment of the imaging polarimetry according to thepresent invention, in the arithmetic step, a value of each element of ageneralized inverse matrix of a matrix is made available such that arelation is formed where a first vector including information on thetwo-dimensional intensity distribution is expressed by a product of thematrix and a second vector including information on the two-dimensionalspatial distribution of the SOP of the light under measurement andinformation on the set of phase attribute functions.

The arithmetic step according to this embodiment is a unit where a valueof each element of the first vector is specified by the use of theobtained intensity distribution, a value of each element of the secondvector is obtained by calculation of a product of the generalizedinverse matrix and the first vector, and by the use of the value of theelement included in the second vector, the set of phase attributefunctions is obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP of the light undermeasurement is obtained.

(11) In another embodiment subject to the embodiment of (10) above,

the set of phase attribute functions is composed of a difference (Δφ₁(x,y)) of the first reference phase function from a reference value forcalibration of the first reference phase function and a difference(Δφ₂(x, y)) of the second reference phase function from a referencevalue for calibration of the second reference phase function. In thearithmetic step of this embodiment, data showing a relation between thefirst reference phase function difference and the second reference phasefunction difference is made available. Further, the generalized inversematrix of the matrix, obtained from the reference value (φ₁ ^((i))(x,y)) for calibration of the first reference phase function and thereference value (φ₂ ^((i))(x, y)) for calibration for the secondreference phase function, is made available.

The arithmetic step is a unit where a value of each element of the firstvector is specified by the use of the obtained intensity distribution, avalue of each element of the second vector is obtained by calculation ofa product of the generalized inverse matrix and the first vector, and bythe use of the value of the element included in the second vector andthe data showing the relation between the first reference phase functiondifference and the second reference phase function difference, the firstreference phase function difference and the second reference phasefunction difference are obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP is obtained.

(12) An imaging polarimeter of the present invention comprises apolarimetric imaging device and an arithmetic unit.

The polarimetric imaging device comprises a first birefringent prismpair, a second birefringent prism pair and an analyzer, through whichlight under measurement passes in sequence, and means for obtaining atwo-dimensional intensity distribution of the light having passedthrough the analyzer are provided, in which each birefringent prism paircomprises parallel flat plates in which two wedge-shaped retardershaving the same apex angle are attached and directions of fast axes ofthe two retarders are orthogonal to each other. Here the secondbirefringent prism pair is arranged such that the direction of aprincipal axis of the second birefringent prism pair disagrees with thedirection of a principal axis of the first birefringent prism pair. Theanalyzer is arranged such that the direction of a transmission axis ofthe analyzer disagrees with the direction of the principal axis of thesecond birefringent prism pair.

In the arithmetic unit, by the use of the two-dimensional intensitydistribution obtained by launching the light under measurement into thepolarimetric imaging device, a set of phase attribute functions of ameasurement system is obtained, and also a parameter indicating atwo-dimensional spatial distribution of a state of polarization (SOP) ofthe light under measurement is obtained. Here, the set of phaseattribute functions is a set of functions defined by properties of thepolarimetric imaging device, and includes a function depending upon atleast a first reference phase function (φ₁(x, y)) as retardation of thefirst birefringent prism pair and a function depending upon at least asecond reference phase function (φ₂(x, y)) as retardation of the secondretarder, and by those functions themselves, or by addition of anotherfunction defined by the properties of the polarimetric imaging device,the set of phase attribute functions becomes a set of functionssufficient to determine a parameter indicating a two-dimensional spatialdistribution of the SOP of the light under measurement.

(13) The analyzer may be arranged such that the direction of thetransmission axis of the analyzer forms an angle of 45° with respect tothe direction of the principal axis of the second birefringent prismpair.

(14) In one embodiment of the imaging polarimeter of the presentinvention, the set of phase attribute functions is composed of the firstreference phase function and the second reference phase function. In thearithmetic unit of this embodiment, data showing a relation between thefirst reference phase function and the second reference phase functionis made available.

The arithmetic unit according to this embodiment is a unit where, by theuse of the two-dimensional intensity distribution obtained by launchingthe light under measurement into the polarimetric imaging device, afirst intensity distribution component which nonperiodically vibrateswith spatial coordinates and a third intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon asecond reference phase function and not depending upon the firstreference phase function are obtained, and at least one of a secondintensity distribution component which vibrates with spatial coordinatesat a frequency depending upon a difference between the first referencephase function and the second reference phase function, a fourthintensity distribution component which vibrates with spatial coordinatesat a frequency depending upon a sum of the first reference phasefunction and the second reference phase function, and a fifth intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon the first reference phase function and notdepending upon the second reference phase function is obtained, and bythe use of the data showing the relation between the first referencephase function and the second reference phase function and each of theobtained intensity distribution components, the first reference phasefunction and the second reference phase function are obtained, and alsothe parameter indicating the two-dimensional spatial distribution of theSOP is obtained.

(15) In another embodiment of the imaging polarimeter of the presentinvention, the set of phase attribute functions is composed of adifference (Δφ₁(x, y)) of the first reference phase function from areference value for calibration of the first reference phase functionand a difference (Δφ₂(x, y)) of the second reference phase function froma reference value for calibration of the second reference phasefunction. In the arithmetic unit of this embodiment, the reference value(φ₁ ^((i))(x, y)) for calibration of the first reference phase function,the reference value (φ₂ ^((i))(x, y)) for calibration of the secondreference phase function, and data showing a relation between the firstreference phase function difference and the second reference phasefunction difference are made available.

The arithmetic unit according to this embodiment is a unit whereby theuse of the two-dimensional intensity distribution obtained by launchingthe light under measurement into the polarimetric imaging device, afirst intensity distribution component which nonperiodically vibrateswith spatial coordinates and a third intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon asecond reference phase function and not depending upon the firstreference phase function are obtained, and at least one of a secondintensity distribution component which vibrates with spatial coordinatesat a frequency depending upon a difference between the first referencephase function and the second reference phase function, a fourthintensity distribution component which vibrates with spatial coordinatesat a frequency depending upon a sum of the first reference phasefunction and the second reference phase function, and a fifth intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon the first reference phase function and notdepending upon the second reference phase function is obtained, and bythe use of the reference value for calibration of the first referencephase function, the reference value for calibration of the secondreference phase function, the data showing the relation between thefirst reference phase function difference and the second reference phasefunction difference, and each of the obtained intensity distributioncomponents, the first reference phase function difference and the secondreference phase function difference are obtained, and also the parameterindicating the two-dimensional spatial distribution of the SOP isobtained.

(16) In another embodiment of the imaging polarimeter according to thepresent invention, in the arithmetic unit, a value of each element of ageneralized inverse matrix of a matrix is made available such that arelation is formed where a first vector including information on thetwo-dimensional intensity distribution is expressed by a product of thematrix and a second vector including information on the two-dimensionalspatial distribution of the SOP of the light under measurement andinformation on the set of the phase attribute function.

The arithmetic unit according to this embodiment is a unit where a valueof each element of the first vector is specified by the use of thetwo-dimensional intensity distribution obtained by launching the lightunder measurement into the polarimetric imaging device, a value of eachelement of the second vector is obtained by calculation of a product ofthe generalized inverse matrix and the first vector, and by the use ofthe value of the element included in the second vector, the set of phaseattribute functions is obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP of the light undermeasurement is obtained.

(17) In another embodiment subject to the embodiment of (16) above, theset of phase attribute functions is composed of a difference (Δφ₁(x, y))of the first reference phase function from a reference value forcalibration of the first reference phase function and a difference(Δφ₂(x, y)) of the second reference phase function from a referencevalue for calibration of the second reference phase function. In thearithmetic unit of this embodiment, data showing a relation between thefirst reference phase function difference and the second reference phasefunction difference is made available. Further, the generalized inversematrix of the matrix, obtained from the reference value (φ₁ ^((i))(x,y)) for calibration of the first reference phase function and thereference value (φ₂ ^((i))(x, y)) for calibration for the secondreference phase function, is made available.

The arithmetic unit in this embodiment is a unit where a value of eachelement of the first vector is specified by the use of thetwo-dimensional intensity distribution obtained by launching the lightunder measurement into the polarimetric imaging device, a value of eachelement of the second vector is obtained by calculation of a product ofthe generalized inverse matrix and the first vector, and by the use ofthe value of the element included in the second vector and the datashowing the relation between the first reference phase functiondifference and the second reference phase function difference, the firstreference phase function difference and the second reference phasefunction difference are obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP is obtained.

According to the present invention, while in principle inheriting theproperty of the imaging polarimetry using the birefringent prism pair inwhich a mechanically movable part for controlling polarized light and anactive element such as an electro-optical modulator are unnecessary, allparameters indicating a two-dimensional spatial distribution of an SOPof light under measurement can be obtained, and it is possible toeffectively reduce a measurement error in the parameter indicating thetwo-dimensional spatial distribution of the SOP, generated by variationsin retardation of a birefringent prism pair due to a temperature changeor other factors.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an explanatory view of a principle of an imagingpolarimetry using a birefringent prism pair as the premise of thepresent invention.

FIG. 2 shows an explanatory view of the birefringent prism pair as thepremise of the present invention.

FIG. 3 shows an explanatory view of a relation between an intensitydistribution obtained from an imaging element and its four components(No. 1).

FIG. 4 shows an explanatory view of a relation between the intensitydistribution obtained from the imaging element and its five components(No. 2).

FIG. 5 shows an explanatory view of a process (flows of signalprocessing) for demodulating a two-dimensional spatial distribution ofStokes parameter.

FIG. 6 shows an explanatory view of one example of Step 2.

FIG. 7 shows an explanatory view of Fourier transformation.

FIG. 8 shows a flowchart of pre-calibration and polarized lightmeasurement.

FIG. 9 shows an explanatory view of flows of signals for calibrationduring measurement.

FIG. 10 shows an explanatory view of flows of signals in the combinationof the “calibration during measurement” and the “measurement of atwo-dimensional spatial distribution of Stokes parameter”.

FIG. 11 shows a comparative explanatory view of methods (No. 1, 2) forcalibrating a reference phase function during measurement.

FIG. 12 shows a constitutional view of one example of an imagingpolarimeter.

FIG. 13 shows a sectional view of the imaging polarimeter shown in FIG.12.

FIG. 14 shows a flowchart of a pre-calibration process.

FIG. 15 shows a flowchart of a measurement process.

FIG. 16 shows a view of an example of experimental results(pre-calibration only)

FIG. 17 shows a view of an example of experimental results (calibrationand calibration during measurement).

FIG. 18 shows a view of an example of experimental results (sectionalview of the measured results)

FIG. 19 shows a constitutional view of an experiment system of animaging polarimetry using a birefringent prism pair proposed by thepresent inventor and the like in advance.

FIG. 20 shows a view of an intensity distribution obtained from animaging element in the same experiment system.

FIG. 21 shows a graph of measured azimuth angle and ellipticity angle inthe same experiment system.

FIG. 22 shows a graph for explaining phase displacement due to atemperature change of the intensity distribution.

FIG. 23 shows a graph for explaining variations in Stokes parameters dueto a temperature change.

FIG. 24 shows a graph for explaining a phase displacement due torelative displacement between the birefringent prism pair and theimaging element.

FIG. 25 shows a view for explaining a relation between the intensitydistribution and a reference phase function.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

A preferred embodiment of the present invention will be described indetail with reference to the accompanying drawings (FIGS. 1 to 11)hereinafter.

Chapter 1: Imaging Polarimetry Using Birefringent Prism Pair as Premiseof the Present Invention

1.1 Principle of Imaging Polarimetry Using Birefringent Prism Pair

FIG. 1 shows a basic constitution of an imaging polarimeter using abirefringent prism pair for use in an imaging polarimetry using thebirefringent prism pair. This imaging polarimeter comprises an imaginglens 102, two birefringent prism pairs 103 (BPP₁ and BPP₂), an analyzerA, and an imaging elements 104. In addition, instead of the imagingelement 104, an object under measurement may be scanned.

As shown in a lower right box in FIG. 1, each birefringent prism pair isconstituted such that a pair of wedge-shaped prisms formed of abirefringent material and having the same apex angle is attached so thattheir oblique sides are alternate. Here, directions of crystal axes ofthe birefringent prisms are perpendicular to an optical axis andintersect with each other at right angles. When this constitution isseen as the whole prism pair, its front and rear surfaces are parallelto each other and are perpendicular to the optical axis, while a contactsurface between the two prisms is slightly inclined with respect to thefront and rear surface of the prism pair. Optically, the birefringentprism pair serves as a retarder in which its property is varieddepending on two-dimensional coordinates. Here, the retarder varies aphase difference between the mutually orthogonal linearly polarizedlight components before and after passage of light through an element.These two intersecting linearly polarized light components are calledprincipal axes and an amount of the phase difference is calledretardation.

The principal axes of the birefringent prism pair BPP₁ and BPP₂ areinclined at 45° from each other and a transmission axis of the analyzerA agrees with one principle axis of the birefringent prism pair BPP₁.

It is to be noted that crossing angles among the three elements (thebirefringent prism pairs BPP₁ and BPP₂ and the analyzer A) may notnecessarily be 45°. Measurement is possible even with a differentcrossing angle, although less efficient to some extent. In short, anycrossing angle can be applied so long as the principal axes of theadjacent elements are not superposed on each other. A description inthis respect-is given later. What is important is that each element isfixed and thus not required to be rotated or modulated as in aconventional method.

In addition, the inclination directions of the contact surfaces of thebirefringent prism pairs BPP₁ and BPP₂ have to be different from eachother. As an example, the above condition is satisfied when the contactsurface of the BPP₂ is inclines only in an x direction and the contactsurface of the BPP₁ is inclined only in a y direction as shown in thelower right box in FIG. 1.

Light under measurement by the polarimeter (light whose state ofpolarization (SOP) is measurable) is light whose SOP varies depending onthe two-dimensional spatial coordinates. According to such light,properties varied in its SOP by transmission, reflection, scattering arecreated by an object under measurement which varies according to thetwo-dimensional spatial coordinates. Two-dimensional spatialdistribution of the SOP of the light under measurement can be expressedby Stokes parameters S₀(x, y), S₁(x, y), S₂ (x, y), and S₃ (x, y)depending on the two-dimensional spatial coordinates (x, y). Further,coordinate axes x and y for determining the Stokes parameters is takenso as to agree with the two intersecting principal axes of the BPP₁.

The light under measurement projecting from the object 101 undermeasurement positioned on the left in the drawing passes in sequencethrough the imaging lens 102, the two birefringent prism pairs 103 (BPP₁and BPP₂), and the analyzer A and is incident on the imaging element104. Here, the imaging lens 102 focuses a projection surface of theobject on an imaging surface of the imaging element 104. In addition, itis assumed that the birefringent prism pairs BPP₁ and BPP₂ and theanalyzer A are sufficiently thin and they are adhered to the imagingsurface of the imaging element 104. Because the imaging surface of theimaging element, on which the projection surface of the object undermeasurement is focused, is to be regarded as the same as the surfaces ofthe prism pairs BPP₁ and BPP₂, that is, spread or blur of image from theprism pair BPP₁ to the imaging element 104 is to be prevented. Inaddition, instead of adhering the birefringent prism pairs BPP, and BPP₂and the analyzer A to the imaging element 104, when relay lenses (secondand third imaging lenses) may be occasionally inserted between the BPP₁and BPP₂ or between the BPP₂ and the imaging element (before or afterthe analyzer A) to keep an imaging relation among the four elements (theprojection surface of the object under measurement, BPP₁, BPP₂, and theinjection surface of the imaging element).

The Stokes parameters depending on the two-dimensional spatialcoordinates x and y are obtained from an intensity distribution acquiredfrom the imaging element by the use of a later-described process.

Before description of the process for obtaining the Stokes parameters,properties of the birefringent prism pairs BPP₁ and BPP₂ are formulatedas a preparation for the process. Here, it is assumed that angles formedbetween the contact surface of the wedges in the birefringent prism pairBPP₁ and the x and y axes are set to γ1x and γ2x. Similarly, inclinationangles γ2x and γ2y of the contact surface in the BPP₂ are specified.Retardation of the birefringent prism pair BPP_(j)(j=1 and 2) can beexpressed by the following expression:φ_(j)(x, y)=2π(U _(jx) x+U _(jy) y)+φ_(j)(x, y)   (1.1)where $\begin{matrix}\left\lbrack {{Mathematical}\quad{expression}{\quad\quad}1} \right\rbrack & \quad \\{U_{jx} = {\frac{2B_{j}}{\lambda}\tan\quad\gamma_{jx}}} & \left( {1.2a} \right) \\{U_{jy} = {\frac{2B_{j}}{\lambda}\tan\quad\gamma_{jy}}} & \left( {1.2b} \right)\end{matrix}$

Here, λ is a wavelength of a light source, and B_(j) is its doublerefraction of a prism medium. Further, φ_(j) (x, y) designates a smallnon-linear component due to imperfectness at the time of processing theprism. As can be seen from the above expressions, the retardation ofeach birefringent prism pair is almost linearly varied with respect tothe spatial coordinates x and y.

In addition, since the inclination of the surfaces in the BPP₁ and BPP₂are different from each other, it is necessary that at least one of thefollowing expressions has to be established.γ1x≠γ2x   (1.3a)γ1y≠γ2y   (1.3b)For example, in case that the contact surface of the BPP₁ is inclined inthe y direction only and the contact surface of the BPP₂ is inclined inthe x direction only (that is, like the lower right box in FIG. 1), thefollowing expressions are provided.γ1x=0   (1.4a)γ1y≠0   (1.4b)γ2x≠0   (1.4c)γ2y=0   (1.4d)

1.2 Intensity Distribution Acquired from Imaging Element

Referring to the “imaging polarimeter using the birefringent prism pair”shown in FIG. 1, the intensity distribution acquired from the imagingelement 104 is expressed by the following expression. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 2} \right\rbrack & \quad \\{{I\left( {x,y} \right)} = {{\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}} + {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}} + {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {\phi_{2}\left( {x,y} \right)} \right\rbrack}} - {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}}} & (1.5)\end{matrix}$whereS ₂₃(x, y)=S ₂(x, y)+iS ₃(x, y)   (1.6)

Here, m₀ (x, y), m⁻(x, y), m₂(x, y), m₊(x, y) each denote a ratio ofamplitude extinction due to failure of imaging element to follow a finevibration component in the intensity distribution. In order tounderstand the property of this expression, Expressions (1, 1) issubstituted into the following Expression (1, 3). $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 3} \right\rbrack & \quad \\{{I\left( {x,y} \right)} = {{\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}} + {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{2{\pi\left( {{U_{- x}x} + {U_{- y}y}} \right)}} + {\Phi_{-}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}} + {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {{2{\pi\left( {{U_{2x}x} + {U_{2y}y}} \right)}} + {\Phi_{2}\left( {x,y} \right)}} \right\rbrack}} - {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{2{\pi\left( {{U_{+ x}x} + {U_{+ y}y}} \right)}} + {\Phi_{+}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}}} & (1.7)\end{matrix}$where the following expressions are satisfied.U _(−x) =U _(2x) −U _(1x)   (1.8a)U _(−y) =U _(2y) −U _(1y)   (1.8b)U _(+x) =U _(2x) +U _(1x)   (1.8c)U _(+y) =U _(2y) +U _(1y)   (1.8d)Φ⁻(x, y)=Φ₂(x, y)−Φ₁(x, y)   (1.8e)Φ₊(x, y)=Φ₂(x, y)+Φ₁(x, y)   (1.8f)

As seen from Expression (1, 7), the intensity distribution I (x, y)acquired from the imaging element contains four components. One of themis a component that gently varies with respect to the spatialcoordinates (x, y), and other three components are quasi-sinusoidalcomponents that vibrate with respect to the spatial coordinates (x, y).These are schematically shown in FIG. 3.

Here, the central spatial frequencies of the three vibration componentsare (U_(−x), U_(−y)), (U_(2x), U_(2y)), (U_(+x), U_(+y)). This meansthat three interference patterns in three different directions aresuperimposed on each other.

What needs to be concerned herein that these four components haveinformation of any one of S₀ (x, y), S₁ (x, y) and S₂₃ (x, y). When eachcomponent can be separated, it is possible to determine thetwo-dimensional spatial distributions S₀ (x, y), S₁ (x, y), S₂ (x, y),and S₃ (X, y) of all Stokes parameters from one intensity distribution I(x, y).

1.3 When Crossing Angle Between Elements is Not 45°

Next described is an intensity distribution acquired in the imagingelement 104 when a crossing angle between the elements is not 45°.

Here also described as a supplemental explanation is an intensitydistribution acquired when a crossing angle between the elements in theoptical system is not 45°.

It is assumed now that, in the optical system shown in FIG. 1, the angleformed between the principal axes of the birefringent prism pairs BPP₁and BPP₂ is θ_(BB) and the angle formed between the principal axis ofthe birefringent prism pairs BPP₂ and the transmission axis of theanalyzer A is θ_(BA). Although calculation has been limited to the caseof θ_(BB)=45° and θ_(BA)=−45°, the case where those angles are morecommon ones is shown here.

An expression for the obtained intensity distribution I (x, y) is givenas follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 4} \right\rbrack & \quad \\{{I\left( {x,y} \right)} = {{\frac{1}{2}{{m_{0}\left( {x,y} \right)}\left\lbrack {{S_{0}\left( {x,y} \right)} + \underset{\_}{\cos\quad 2\theta_{BA}\cos\quad 2\theta_{BB}{S_{1}\left( {x,y} \right)}}} \right\rbrack}} - {\frac{1}{2}\left( {\sin\quad 2\theta_{BA}\sin^{2}\theta_{BB}} \right){m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}} - {\frac{1}{2}\left( {\sin\quad 2\theta_{BA}\sin\quad 2\theta_{BB}} \right){m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {\phi_{2}\left( {x,y} \right)} \right\rbrack}} + {\frac{1}{2}\left( {\sin\quad 2\theta_{BA}\cos^{2}\theta_{BB}} \right){m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}} + \underset{\_}{\frac{1}{2}\left( {\cos\quad 2\theta_{BA}\sin\quad 2\theta_{BB}} \right){m_{1}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}}} & (1.9)\end{matrix}$

These are schematically shown in FIG. 4.

When this expression is compared with the intensity distribution in theprevious expression (1.5), namely when the angles θ_(BB) and θ_(BA) arerespectively limited to 45° and −45°, the following differences arefound in addition to a mere difference in constant multiple of acoefficient. It is to be noted that the different part is indicated withan underline in Expression (1. 9).

The component that gently varies with respect to the intensitydistribution (x, y) depends not only upon S₀ (x, y) but additionallyupon S₁ (x, y).

A component that quasi-sinusoidally vibrates according to the phase φ₁(x, y), namely a component that vibrates at a central spatial frequency(U_(1x), U_(1y)) is added. It should be noted that this component hasinformation of S₂₃ (x, y) like the two components which vibrateaccording to the φ₂ (x, y)−φ₁ (x, y) and φ₂ (x, y)+φ₁ (x, y). It meansthat this term can be treated in the same manner as the other two termsincluding S₂₃.

Here, conditions for nonappearance of the above two components areconsidered.

The former term appears in a limited case “when both θ_(BB)≠±45° andθ_(BA)≠±45° are satisfied”. Meanwhile, the latter term appears “whenθ_(BA)≠±45° (regardless of whether θ_(BB) aggress with 45° or not)”.From these, a fact can be mentioned as follows.

When the principal axis of the birefringent prism pair BPP₂ and thetransmission axis of the analyzer A cross each other at an angle of 45°(i.e. θ_(BA)≠±45°), the intensity distribution obtained from the imagingelement is given by Expression (1. 5) except for the difference inconstant multiple of a coefficient of each term. Here, whether the angleθ_(BB) formed between the principal axes of the birefringent prism pairsBPP₁ and BPP₂ agrees with ±45° or not is irrelevant.

In other words, the intensity distribution can take the form ofExpression (1.5) under a condition that the principal axis of thebirefringent prism pair BPP₂ and the transmission axis of the analyzer Across each other at an angle of 45°. On the other hand, whether theangle formed between the principal axes of the birefringent prism pairsBPP₁ and BPP2 agrees with ±45° or not is irrelevant.

1.4 Process for Demodulating Two-Dimensional Spatial Distribution ofStokes Parameter

A specific process for demodulating the two-dimensional spatialdistribution of the stokes parameter is described below with referenceto FIG. 5. A brief description of the flow of the process is as follows.

Step 1: Each term is separated from the intensity distribution I (x, y)obtained from the imaging element.

Step 2: An amplitude and a phase of each component are obtained. (Orequivalent quantities, e.g. a real part and an imaginary part in complexrepresentation are obtained).

Step 3:${\left\lbrack {{Mathematical}\quad{Expression}\quad 5} \right\rbrack \cdot {reference}}\quad{amplitude}\quad{function}\quad{\begin{Bmatrix}{m_{0}\left( {x,y} \right)} \\{m_{-}\left( {x,y} \right)} \\{m_{2}\left( {x,y} \right)} \\{m_{+}\left( {x,y} \right)}\end{Bmatrix} \cdot {reference}}\quad{phase}\quad{function}\quad\begin{Bmatrix}{\phi_{1}\left( {x,y} \right)} \\{\phi_{2}\left( {x,y} \right)}\end{Bmatrix}$

The above reference functions included in an amplitude and a phase ofeach vibration component are removed to obtain the two-dimensionalspatial distributions S₀ (x, y), S₁ (x, y), S₂ (x, y) and S₃ (X, y) ofthe Stokes parameters. (These reference functions depend not upon lightunder measurement but only upon parameters of a polarimeter).

Each of the steps is described as follows.

[Step 1]

As described in the previous section, the intensity distribution I (x,y) obtained from the imaging element contains four components. Anoperation for taking out each component by a signal process isperformed. What is applied to this operation is that each componentvibrates at a different period (frequency). With the use of (any one ona variety of frequency filtering techniques being broadly used in fieldsof communication engineering, signal analysis and the like, it ispossible to separate each component. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 6} \right\rbrack & \quad \\{{{\cdot {{component}{\quad\quad}\lbrack 1\rbrack}}\left( {{low}\quad{frequency}\quad{component}} \right)}{\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}} & \left( {1.10a} \right) \\{\left. {{\cdot {{component}\quad\lbrack 2\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right){\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}} & \left( {1.10b} \right) \\{\left. {{\cdot {{component}\quad\lbrack 3\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)} \right){\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {\phi_{2}\left( {x,y} \right)} \right\rbrack}}} & \left( {1.10c} \right) \\{\left. {{\cdot {{component}\quad\lbrack 4\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right) - {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}} & \left( {1.10d} \right)\end{matrix}$

component [1] is a component of intensity distribution whichnonperiodically vibrates with spatial coordinates component [2] is acomponent of intensity distribution which vibrates with spatialcoordinates at a frequency depending upon a difference between the firstreference phase function φ₁(x, y) and the second reference phasefunction φ₂(x, y) component [3] is a component of intensity distributionwhich vibrates with spatial coordinates at a frequency depending upon asecond reference phase function φ₂(x, y) and not depending upon thefirst reference phase function φ₁(x, y) component [4] is a component ofintensity distribution which vibrates with spatial coordinates at afrequency depending upon a sum of the first reference phase functionφ₁(x, y) and the second reference phase function φ₂(x, y). When theprincipal axis of the birefringent prism pair BPP₂ and the transmissionaxis of the analyzer A is not 45°, a component of intensity distribution[5] appears which vibrates with special coordinates at a frequencydepending upon the first reference phase function φ₁(x, y) and notdepending upon the second reference phase function φ₂(x, y).

[Step 2]

As for each component separated in Step 1, a “set of an amplitude and aphase” and a “complex representation” are obtained, as shown in FIG. 6.This can be readily realized by using a variety of demodulation methodswhich are common in fields of communication engineering, signal analysisand the like, as in Step 1. Those methods include:

Amplitude demodulation: rectifying demodulation, envelope demodulation,etc.

Phase demodulation: frequency discrimination, zero-crossing method, etc.

Complex representation demodulation: Fourier transform method (laterdescribed), synchronous demodulation, etc.

Here, definitions and basic properties of the “amplitude”, “phase” and“complex representation” of a vibration component are summarized below.As seen from Expressions (1.10a) to (1.10d), each of the separatedcomponents except for component [1 ] takes the form of:a(x, y)cos δ(x, y)   (1.11)a(x, y) and δ (x, y) here are respectively referred to as the“amplitude” and “phase” of the vibration component. It is to be notedthat, if assuming that the phase δ₀(x, y)=0 (i.e. cos δ₀(x, y)=1) alsoin component [1], the amplitude of this component can also be defined.

Further, F(x, y) having the following relation with the amplitude andthe phase is called a complex representation. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 7} \right\rbrack & \quad \\\begin{matrix}{{F\left( {x,y} \right)} = {\frac{1}{2}{a\left( {x,y} \right)}{\exp\left\lbrack {i\quad{\delta\left( {x,y} \right)}} \right\rbrack}}} \\{= {\left\lbrack {\frac{1}{2}{a\left( {x,y} \right)}\cos\quad{\delta\left( {x,y} \right)}} \right\rbrack + {i\left\lbrack {\frac{1}{2}{a\left( {x,y} \right)}\sin\quad{\delta\left( {x,y} \right)}} \right\rbrack}}}\end{matrix} & \begin{matrix}\left( {1.12a} \right) \\\left( {1.12b} \right)\end{matrix}\end{matrix}$The real part of F(x, y) is formed by dividing the amplitude of thevibration component into halves, and the imaginary part thereof isdisplaced from the real part at the angle of 90°. It should be notedthat in component [1], the amplitude is not divided into halves since δ(x, y)=0, i.e. no imaginary part exists.

What needs to be concerned here is that when either the “set of theamplitude and phase” or the “complex representation” is demodulated, theother one can be immediately calculated by the use of the followingrelational expression. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 8} \right\rbrack & \quad \\{{{``{{{amplitude}\quad{a\left( {x,y} \right)}},{{phase}\quad{\delta\left( {x,y} \right)}}}"}->{``{{complex}\quad{representation}{\quad\quad}{F\left( {x,y} \right)}}"}}{{F\left( {x,y} \right)} = {\frac{1}{2}{a\left( {x,y} \right)}{\mathbb{e}}^{{\mathbb{i}\delta}{({x,y})}}}}} & (1.13) \\{{{``{{complex}\quad{representation}{\quad\quad}{F\left( {x,y} \right)}}"}->{``{{{amplitude}\quad{a\left( {x,y} \right)}},{{phase}\quad{\delta\left( {x,y} \right)}}}"}}{{a\left( {x,y} \right)} = {2{{F\left( {x,y} \right)}}}}} & \left( {1.14a} \right) \\{{\delta\left( {x,y} \right)} = {\arg\left\lbrack {F\left( {x,y} \right)} \right\rbrack}} & \left( {1.14b} \right)\end{matrix}$

Namely, when on is demodulated, the other can be immediately calculatedas necessary.

When the “amplitude” and “phase” of each component are demodulated, thefollowing results are obtained. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 9} \right\rbrack & \quad \\{{{\cdot {{component}\quad\lbrack 1\rbrack}}\left( {{low}\quad{frequency}\quad{component}} \right)}\begin{matrix}{``{amplitude}"} & {{a_{0}\left( {x,y} \right)} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}} \\{``{phase}"} & {{\delta_{0}\left( {x,y} \right)} = 0}\end{matrix}} & \left( {1.15a} \right) \\{\left. {{\cdot {{component}\quad\lbrack 2\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{-}\left( {x,y} \right)} = {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}}} \\{``{phase}"} & {{\delta_{-}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}}}\end{matrix}} & \left( {1.15b} \right) \\{\left. {{\cdot {{component}\quad\lbrack 3\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{2}\left( {x,y} \right)} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}}} \\{``{phase}"} & {{\delta_{2}\left( {x,y} \right)} = {\phi_{2}\left( {x,y} \right)}}\end{matrix}} & \left( {1.15c} \right) \\{\left. {{\cdot {{component}\quad\lbrack 4\rbrack}}\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{+}\left( {x,y} \right)} = {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}}} \\{``{phase}"} & {{\delta_{+}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}} + \pi}}\end{matrix}} & \left( {1.15d} \right)\end{matrix}$

On the other hand, when the “complex representation” of each componentis demodulated, the following results are obtained. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 10} \right\rbrack & \quad \\{{{{component}\quad\lbrack 1\rbrack}\quad\left( {{low}\quad{frequency}\quad{component}} \right)}{{{``{{complex}\quad{representation}}"}\quad{F_{0}\left( {x,y} \right)}} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}}} & \left( {1.16a} \right) \\{\left. {{{component}\quad\lbrack 2\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{-}\left( {x,y} \right)}} = {\frac{1}{8}{m_{-}\left( {x,y} \right)}{S_{23}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\quad\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}} \right\rbrack}}}} & \left( {1.16b} \right) \\{\left. {{{component}\quad\lbrack 3\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{2}\left( {x,y} \right)}} = {\frac{1}{4}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}\exp\quad{{\mathbb{i}\phi}_{2}\left( {x,y} \right)}}}} & \left( {1.16c} \right) \\{\left. {{{component}\quad\lbrack 4\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{+}\left( {x,y} \right)}} = {{- \frac{1}{8}}{m_{+}\left( {x,y} \right)}{S_{23}^{*}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}} \right\rbrack}}}} & \left( {1.16d} \right)\end{matrix}$Here, * denotes a complex conjugation. It is to be noted that, for thesake of what is described below, the expressions of the complexrepresentations are rewritten as follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 11} \right\rbrack & \quad \\{{{{component}\quad\lbrack 1\rbrack}\quad\left( {{low}\quad{frequency}\quad{component}} \right)}{{{``{{complex}\quad{representation}}"}\quad{F_{0}\left( {x,y} \right)}} = {{K_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}}} & \left( {1.17a} \right) \\{\left. {{{component}\quad\lbrack 2\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{-}\left( {x,y} \right)}} = {{K_{\_}\left( {x,y} \right)}{S_{23}\left( {x,y} \right)}}}} & \left( {1.17b} \right) \\{\left. {{{component}\quad\lbrack 3\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{2}\left( {x,y} \right)}} = {{K_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}}}} & \left( {1.17c} \right) \\{\left. {{{component}\quad\lbrack 4\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right){{{``{{complex}\quad{representation}}"}\quad{F_{+}\left( {x,y} \right)}} = {{K_{+}\left( {x,y} \right)}{S_{23}^{*}\left( {x,y} \right)}}}} & \left( {1.17d} \right) \\{where} & \quad \\{{K_{0}\left( {x,y} \right)} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}}} & \left( {1.18a} \right) \\{{K_{-}\left( {x,y} \right)} = {\frac{1}{8}{m_{-}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}} \right\rbrack}}} & \left( {1.18b} \right) \\{{K_{2}\left( {x,y} \right)} = {\frac{1}{4}{m_{2}\left( {x,y} \right)}\exp\quad{{\mathbb{i}\phi}_{2}\left( {x,y} \right)}}} & \left( {1.18c} \right) \\{{K_{+}\left( {x,y} \right)} = {{- \frac{1}{8}}{m_{+}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}} \right\rbrack}}} & \left( {1.18d} \right)\end{matrix}$[Step 3]

Finally, from the “amplitude” and the “phase” or the “complexrepresentation” obtained in Step 2, the two-dimensional spatialdistributions S₀(x, y), S₁(x, y), S₂(x, y), and S₃(x, y) of the Stokesparameters as functions of the spatial coordinates (x, y) aredetermined.

The “amplitude” and the “phase” obtained in Step 2 include, other thanthe two-dimensional spatial distribution of the Stokes parameters to beobtained, parameters shown below. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 12} \right\rbrack & \quad \\{{{Parameter}\quad({function})\quad{determined}\quad{based}\quad{only}\quad{upon}}{{property}\quad{of}\quad{polarimeter}}\begin{Bmatrix}{m_{0}\left( {x,y} \right)} \\{m_{-}\left( {x,y} \right)} \\{m_{2}\left( {x,y} \right)} \\{m_{+}\left( {x,y} \right)}\end{Bmatrix}{and}\begin{Bmatrix}{{\phi_{-}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}}} \\{\phi_{2}\left( {x,y} \right)} \\{{\phi_{+}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}}}\end{Bmatrix}} & \quad\end{matrix}$

The former are included in the amplitude while the latter are includedin the phase. These provide references in determining thetwo-dimensional spatial distribution of the Stokes parameters from theamplitude and the phase of each vibration component. Thus, each of thesefunctions is hereinafter referred to as a “reference amplitude function”and a “reference phase function”. Since these parameters do not dependupon light under measurement, each of the parameters is subjected todivision or subtraction, to be determined as follows.

-   -   S₀(x, y) can be determined from [component [1]].    -   S₂(x, y) and S₃(x, y) can be determined from (either) [component        [2]] or [component [4]].    -   S₁(x, y) can be determined from [component [3]].

Meanwhile, in the case of the “complex representation”, parameters(functions) determined only by the property of the polarimeter itselfare K₀(x, y), K⁻(x, y), K₂(x, y), and K₊(x, y) which are defined byExpressions (1.18a) to (1.18d). These are, so to speak, “referencecomplex functions”.

As revealed from Expressions (1.17a) to (1.17d), if the above referencecomplex functions have been obtained, by division of the complexrepresentation of each vibration component demodulated in Step 2, theparameters can be determined as follows.

-   -   S₀(x, y) can be determined from [component [1]]    -   S₂(x, y) and S₃(x, y) can be determined from (either) [component        [2]] or [component [4]].    -   S₁(x, y) can be determined from [component [3]].

When the angle formed between the birefringent prism pair BPP₂ and theanalyzer A is not 45°, the fifth term that appears can be used in placeof [component [2]] and [component [4]]. Namely, the description on line2 above can be rewritten to:

-   -   S₂(x, y) and S₃(x, y) can be determined from any one of        [component [2]], [component [4]] and [component [5]].

Next, as one of signal processing methods for demodulating thetwo-dimensional spatial distribution of the Stokes parameters, a“Fourier transform method” is described with reference to FIG. 7. Theuse of this method allows efficient concurrent performance of Steps 1and 2, leading to immediate determination of all complex representationsof each vibration component.

In this method, first, the intensity distribution I(x, y) measured withthe imaging element in the imaging polarimeter using the birefringentprism pair is subjected to Fourier transformation, to obtain thefollowing two-dimensional spatial-frequency spectrum. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 13} \right\rbrack & \quad \\{{\overset{\sim}{I}\left( {f_{x},f_{y}} \right)} = {{A_{0}\left( {f_{x},f_{y}} \right)} + {A_{-}\left( {{f_{x} - U_{- x}},{f_{y} - U_{- y}}} \right)} + {A_{-}^{*}\left( {{{- f_{x}} - U_{- x}},{{- f_{y}} - U_{- y}}} \right)} + {A_{2}\left( {{f_{x} - U_{2x}},{f_{y} - U_{2y}}} \right)} + {A_{2}^{*}\left( {{{- f_{x}} - U_{2x}},{{- f_{y}} - U_{2y}}} \right)} + {A_{+}\left( {{f_{x} - U_{+ x}},{f_{y} - U_{+ y}}} \right)} + {A_{+}^{*}\left( {{{- f_{x}} - U_{+ x}},{{- f_{y}} - U_{+ y}}} \right)}}} & (1.19) \\{where} & \quad \\{{A_{0}\left( {f_{x},f_{y}} \right)} = {F^{- 1}\left\lbrack {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}} \right\rbrack}} & \left( {1.20\quad a} \right) \\{{A_{-}\left( {f_{x},f_{y}} \right)} = {F^{- 1}\left\lbrack {\frac{1}{8}{m_{-}\left( {x,y} \right)}{S_{23}\left( {x,y} \right)}\exp\quad{\mathbb{i}}\quad{\Phi_{-}\left( {x,y} \right)}} \right\rbrack}} & \left( {1.20b} \right) \\{{A_{2}\left( {f_{x},f_{y}} \right)} = {F^{- 1}\left\lbrack {\frac{1}{4}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}\exp\quad{\mathbb{i}}\quad{\Phi_{2}\left( {x,y} \right)}} \right\rbrack}} & \left( {1.20c} \right) \\{{A_{+}\left( {f_{x},f_{y}} \right)} = {F^{- 1}\left\lbrack {{- \frac{1}{8}}{m_{+}\left( {x,y} \right)}{S_{23}^{*}\left( {x,y} \right)}\exp\quad{\mathbb{i}}\quad{\Phi_{+}\left( {x,y} \right)}} \right\rbrack}} & \left( {1.20d} \right)\end{matrix}$is spatial-frequency spectrum I(f_(x), f_(x)) contains seven components.In addition, as shown in the lower right box in FIG. 1, when theinclination of the contact surfaces of the BPP₁ and BPP₂ are only in they direction and only in the x direction, respectively, the abovespectrum becomes as follows. $\begin{matrix}{{\overset{\sim}{I}\left( {f_{x},f_{y}} \right)} = {{A_{0}\left( {f_{x},f_{y}} \right)} + {A_{-}\left( {{f_{x} - U_{2x}},{f_{y} - U_{1y}}} \right)} + {A_{-}^{*}\left( {{{- f_{x}} - U_{2x}},{{- f_{y}} - U_{1y}}} \right)} + {A_{2}\left( {{f_{x} - U_{2x}},f_{y}} \right)} + {A_{2}^{*}\left( {{{- f_{x}} - U_{2x}},{- f_{y}}} \right)} + {A_{+}\left( {{f_{x} - U_{2x}},{f_{y} - U_{1y}}} \right)} + {A_{+}^{*}\left( {{{- f_{x}} - U_{2x}},{{- f_{y}} - U_{1y}}} \right)}}} & (1.21)\end{matrix}$This spectrum is schematically shown in a upper left box in FIG. 7.

Meanwhile, the central special frequency of the seven componentscontained in the spatial-frequency spectrum I(f_(x), f_(x)) are (0,0),±(U_(−x), U_(−y)), ±(U_(2x), U_(2y)), and ±(U_(+x), U_(+y)). Here, whenappropriate selection of these inverse frequencies, the componentscontained in the Ĩ(f_(x), f_(x)) can be separated in the two-dimensionalspatial-frequency space from each other.

When four components with (f_(x), f_(x))=(0,0), (U_(−x), U_(−y)),(U_(2x), U_(2y)), and (U_(+x), U_(+y)) at the nucleus are taken out andthen subjected to the Fourier transformation, the following expressionsare satisfied. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 14} \right\rbrack & \quad \\\begin{matrix}{{F\left\lbrack {A_{0}\left( {f_{x},f_{y}} \right)} \right\rbrack} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}} \\{= {F_{0}\left( {x,y} \right)}}\end{matrix} & \left( {1.22a} \right) \\\begin{matrix}{{F\left\lbrack {A_{-}\left( {{f_{x} - U_{- x}},{f_{y} - U_{- y}}} \right)} \right\rbrack} = {\frac{1}{8}{m_{-}\left( {x,y} \right)}{S_{23}\left( {x,y} \right)}\exp\quad{\mathbb{i}}}} \\{\left\lbrack {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}} \right\rbrack} \\{= {F_{-}\left( {x,y} \right)}}\end{matrix} & \left( {1.22b} \right) \\\begin{matrix}{{F\left\lbrack {A_{2}\left( {{f_{x} - U_{2x}},{f_{y} - U_{2y}}} \right)} \right\rbrack} = {\frac{1}{4}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}\exp\quad{{\mathbb{i}\phi}_{2}\left( {x,y} \right)}}} \\{= {F_{2}\left( {x,y} \right)}}\end{matrix} & \left( {1.22c} \right) \\\begin{matrix}{{F\left\lbrack {A_{+}\left( {{f_{x} - U_{+ x}},{f_{y} - U_{+ y}}} \right)} \right\rbrack} = {{- \frac{1}{8}}{m_{+}\left( {x,y} \right)}{S_{23}^{*}\left( {x,y} \right)}\exp\quad{\mathbb{i}}}} \\{\left\lbrack {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}} \right\rbrack} \\{= {F_{+}\left( {x,y} \right)}}\end{matrix} & \left( {1.22d} \right)\end{matrix}$

As seen from the expressions above, what are obtained in the aboveoperation are just the complex representations of the components [1] to[4] to be obtained in foregoing Step 2. Namely, in the above operations,Steps 1 and 2 are concurrently realized. Hence, when Step 3 is performedusing the results of Steps 1 and 2, all two-dimensional spatialdistribution of the Stokes parameters is obtained all at once.

1.5 Pre-Calibration: Calibration of Reference Amplitude Function,Reference Phase Function, Reference Complex Function “Prior toMeasurement”

As described in the previous section, when the two-dimensional spatialdistribution (two-dimensional spatial distribution of the Stokesparameters) of SOP of the light under measurement is determined from theintensity distribution obtained from the imaging element, it isnecessary to determine in advance in Step 3 parameters to be obtainedbased only on a property of the polarimeter itself, namely:

“reference amplitude function” m₀(x, y), m⁻(x, y), m₂(x, y), m₊(x, y),and “reference phase function” φ₂(x, y) and φ₁(x, y), or

“reference complex function” K₀(x, y), K⁻(x, y), K₂(x, y), K₊( x, y).

The former (“reference amplitude function” and “reference phasefunction”) and the latter (“reference complex function”) are required inthe respective cases of obtaining the two-dimensional spatialdistribution of the Stokes parameters from the “amplifier and phase” orthe “complex representation” of each vibration component. Since theseare functions not depending upon the light under measurement, it isdesirable to calibrate the functions at least prior to measurement.

In this section, a process for calibrating these reference functions“prior to measurement, i.e. in advance” is described. Namely, as shownin FIG. 8, pre-calibration (Steps 701 to 705) needs to be performedprior to polarization measurement (Steps 711 to 714). There are twotypical methods as follows.

[Method 1]: a method for calibrating reference phase functions andreference amplitude functions based upon a property of each element foruse in the optical system.

[Method 2]: a method for calibrating reference phase functions andreference amplitude functions by the use of light having a known SOP.

1.5.1 [Method 1]

Method for Calibrating Reference Phase Function and Reference AmplitudeFunction Based Upon Property of Each Element for Use in Optical System

Properties of a reference phase function and a reference amplitudefunction are essentially determined based upon elements for use in animaging polarimeter using a birefringent prism pair. Therefore, opticalproperties of individual elements are repeatedly examined by experimentor calculation to perform calibration of parameters.

1.5.2 [Method 2]

Method for Calibrating Reference Phase Function and Reference AmplitudeFunction by Use of Light having a Known SOP

The reference phase function and the reference amplitude function are inamount determined based not upon “SOP of light under measurement”, butonly upon the property of the “imaging polarimeter using thebirefringent prism pair”. Accordingly, the “light having a known SOP”(light whose measurement result is known)“is inputted into thepolarimeter, and using the result of the input, it is possible tocalculate backward the reference phase function and the referenceamplitude function.

It is to be noted that the “imaging polarimeter using the birefringentprism pair” has the following advantages.

The “light whose SOP is known” may be “only one kind” of light.

“Linearly polarized light” can be used as the “only one kind of light”.

In a currently used polarimeter for obtaining the two-dimensionalspatial distribution of the Stokes parameters, it has been normallyrequired in calibration that at least four kinds of light with differentstates of polarization be prepared and further that at least one kind oflight be not linearly polarized light. As opposed to this, in theimaging polarimeter using the birefringent prism pair, only one kind ofknown polarized light is required, and it may further be linearlypolarized light. The linearly polarized light is convenient because,unlike light in other SOPs, the linearly polarized light can facilitatecreation of precisely controlled polarized light by means ofhigh-extinction polarizer made of crystal.

Below, the process for calibration is shown. As described at thebeginning of this section, the following should be noted.

When the SOP is obtained from the “amplitude and phase” of eachvibration component, the “reference amplitude function” and the“reference phase function” are required.

When the SOP is obtained from the “complex representation” of eachvibration component, the “reference complex function” is required.

In the following, the respective processes for calibration in the abovetwo cases are described. Although these processes are essentiallyequivalent and different only in calculation method, they are separatelyput down for the sake of convenience.

A. Calibration Process for Separately Obtaining Reference AmplitudeFunction and Reference Phase Function

In this calibration, first, “light having some known SOP” is prepared,and then incident on an imaging polarimeter using a birefringent prismpair. Two-dimensional spatial distribution of the Stokes parameters ofthe known light are referred to as S₀ ⁽⁰⁾(x, y), S₁ ⁽⁰⁾(x, y), S₂ ⁽⁰⁾(x,y), and S₃ ⁽⁰⁾(x, y). When the light is subjected to the above-mentioneddemodulation means, the amplitude and the phase obtained in Step 2 areexpressed as follows according to Expressions (1.15a) to (1.15d).$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 15} \right\rbrack & \quad \\{{{{component}\quad\lbrack 1\rbrack}\quad\left( {{low}\quad{frequency}\quad{component}} \right)}{{{``{amplitude}"}\quad{a_{0}^{(0)}\left( {x,y} \right)}} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}^{(0)}\left( {x,y} \right)}}}{{{``{phase}"}\quad{\delta_{0}^{(0)}\left( {x,y} \right)}} = 0}} & \left( {1.23a} \right) \\{\left. {{{component}\quad\lbrack 2\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right){{{``{amplitude}"}\quad{a_{-}^{(0)}\left( {x,y} \right)}} = {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}^{(0)}\left( {x,y} \right)}}}}{{{``{phase}"}\quad{\delta_{-}^{(0)}\left( {x,y} \right)}} = {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}^{(0)}\left( {x,y} \right)} \right\}}}}} & \left( {1.23b} \right) \\{{{{component}\quad\lbrack 3\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)}{{{``{amplitude}"}\quad{a_{2}^{(0)}\left( {x,y} \right)}} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}^{(0)}\left( {x,y} \right)}}}{{{``{phase}"}\quad{\delta_{2}^{(0)}\left( {x,y} \right)}} = {\phi_{2}\left( {x,y} \right)}}} & \left( {1.23c} \right) \\{\left. {{{component}\quad\lbrack 4\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right){{{``{amplitude}"}\quad{a_{+}^{(0)}\left( {x,y} \right)}} = {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}^{(0)}\left( {x,y} \right)}}}}{{{``{phase}"}\quad{\delta_{+}^{(0)}\left( {x,y} \right)}} = {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {{S_{23}^{(0)}\left( {x,y} \right)} + \pi} \right.}}}} & \left( {1.23d} \right) \\{{where},} & \quad \\{{S_{23}^{(0)}\left( {x,y} \right)} = {{S_{2}^{(0)}\left( {x,y} \right)} + {{iS}_{3}^{(0)}\left( {x,y} \right)}}} & \left( 1.24 \right.\end{matrix}$It is to be noted that this is mere replacement of S₀(x, y) to S₃(x, y)with S₀ ⁽⁰⁾(x, y) to S₃ ⁽⁰⁾(x, y).

The phase and the amplitude of each vibration component are determinedonly by the two-dimensional spatial distribution of the Stokesparameters, the reference phase functions and the reference amplitudefunctions. Here, since the two-dimensional spatial distribution of theStokes parameters are known in a “case where light whose SOP is known isincident”, the remaining reference amplitude functions m₀(x, y), m⁻(x,y), m₂(x, y), m₊( x, y), and reference phase functions φ₁(x, y) andφ₂(x, y) are determined from the demodulated amplitude and phase.Specifically, these functions are given according to the followingexpressions: $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 16} \right\rbrack & \quad \\{{m_{0}\left( {x,y} \right)} = \frac{2{a_{0}^{(0)}\left( {x,y} \right)}}{S_{0}^{(0)}\left( {x,y} \right)}} & \left( {1.25a} \right) \\{{m_{-}\left( {x,y} \right)} = \frac{4{a_{-}^{(0)}\left( {x,y} \right)}}{{S_{23}^{(0)}\left( {x,y} \right)}}} & \left( {1.25b} \right) \\{{m_{2}\left( {x,y} \right)} = \frac{2{a_{2}^{(0)}\left( {x,y} \right)}}{S_{1}^{(0)}\left( {x,y} \right)}} & \left( {1.25c} \right) \\{{m_{+}\left( {x,y} \right)} = \frac{4{a_{+}^{(0)}\left( {x,y} \right)}}{{S_{23}^{(0)}\left( {x,y} \right)}}} & \left( {1.25d} \right) \\\begin{matrix}{{\phi_{-}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}}} \\{= {{\delta_{-}^{(0)}\left( {x,y} \right)} - {\arg\left\{ {S_{23}^{(0)}\left( {x,y} \right)} \right\}}}}\end{matrix} & \left( {1.25e} \right) \\{{\phi_{2}\left( {x,y} \right)} = {\delta_{2}^{(0)}\left( {x,y} \right)}} & \left( {1.25f} \right) \\\begin{matrix}{{\phi_{+}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}}} \\{= {{\delta_{+}^{(0)}\left( {x,y} \right)} + {\arg\left\{ {S_{23}^{(0)}\left( {x,y} \right)} \right\}} - \pi}}\end{matrix} & \left( {1.25g} \right)\end{matrix}$Once these reference functions are obtained (can be calibrated),two-dimensional spatial distribution of the Stokes parameters of thelight having an unknown SOP can be obtained.

It should be noted that it is seen from the above that the condition forthe light having a known SOP is only that S₀ ⁽⁰⁾(x, y), S₁ ⁽⁰⁾(x, y) andS₂₃ ⁽⁰⁾(x, y) are not zero. In particular, as for the last S₂₃ ⁽⁰⁾(x,y), it is meant that the condition is satisfied even when one of S₂⁽⁰⁾(x, y) and S₃(o)(x, y) is zero if the other is not zero. Here, S₃⁽⁰⁾(x, y)=0 means linearly polarized light. Namely, calibration ispossible by the use of linearly polarized light alone. Specifically,when linearly polarized light with an azimuth 0 is used as the knownlight, those are expressed as follows.S ₀ ⁽⁰⁾(x, y)=I ⁽⁰⁾(x, y)   (1.26a)S ₁ ⁽⁰⁾(x, y)=I ⁽⁰⁾(x, y)cos 2θ  (1.26b)S ₂ ⁽⁰⁾(x, y)=I ⁽⁰⁾(x, y)sin 2θ  (1.26c)S ₃ ⁽⁰⁾(x, y)=0   (1.26d)Here, I₀ ⁽⁰⁾(x, y) is an intensity distribution of calibration light. Inthis case, the above expressions (1.25a) to (1.25g) are expressed asfollows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 17} \right\rbrack & \quad \\{{m_{0}\left( {x,y} \right)} = \frac{2{a_{0}^{(0)}\left( {x,y} \right)}}{I^{(0)}\left( {x,y} \right)}} & \left( {1.27a} \right) \\{{m_{-}\left( {x,y} \right)} = \frac{4{a_{-}^{(0)}\left( {x,y} \right)}}{{I^{(0)}\left( {x,y} \right)}\sin\quad 2\theta}} & \left( {1.27b} \right) \\{{m_{2}\left( {x,y} \right)} = \frac{2{a_{2}^{(0)}\left( {x,y} \right)}}{{I^{(0)}\left( {x,y} \right)}\cos\quad 2\theta}} & \left( {1.27c} \right) \\{{m_{+}\left( {x,y} \right)} = \frac{4{a_{+}^{(0)}\left( {x,y} \right)}}{{I^{(0)}\left( {x,y} \right)}\sin\quad 2\theta}} & \left( {1.27d} \right) \\{{\phi_{-}\left( {x,y} \right)} = {{{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)}} = {\delta_{-}^{(0)}\left( {x,y} \right)}}} & \left( {1.27e} \right) \\{{\phi_{2}\left( {x,y} \right)} = {\delta_{2}^{(0)}\left( {x,y} \right)}} & \left( {1.27f} \right) \\{{\phi_{+}\left( {x,y} \right)} = {{{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)}} = {{\delta_{+}^{(0)}\left( {x,y} \right)} - \pi}}} & \left( {1.27g} \right)\end{matrix}$

It is revealed from the above that the reference amplitude function andthe reference phase function can be obtained if only the azimuth angle θand the intensity distribution I⁽⁰⁾(x, y) of a light source are known inadvance. Further, even with I⁽⁰⁾(x, y) unknown, if only the azimuthangle θ is known, it can still be sufficient for use in obtaining partof (essential) polarized light parameters.

B. Calibration Process for Obtaining Both Altogether (By Regarding Bothas Reference Complex Function) at Once

The above-mentioned method was a method for calculating the “amplitude”and the “phase” of each vibration component separately. However, it maybe more convenient (efficient) in some cases to calculate them as the“complex representation” of each vibration component. One example ofsuch calculation may be the case of directly obtaining the “complexrepresentation” (Expressions (1.17a) to (1.17d)), as in the Fouriertransform method shown in FIG. 7 above. In such a case, calibration isefficiently performed when the “complex representation” is calibrated asit is without separation into the “amplitude” and “phase”.

In the following, mathematical expressions for the above-mentioned caseare shown. What needs to be concerned here is that the physical naturesof the cases of using “amplitude and phase” and the “complexrepresentation” are completely the same. It is just that in the lattercase, a calculation is made using complex numbers, and thus moreefficient.

Similarly to the previous section, a case is considered where lighthaving known two-dimensional spatial distributions S₀ ⁽⁰⁾(x, y), S₁⁽⁰⁾(x, y), S₂ ⁽⁰⁾(x, y), and S₃ ⁽⁰⁾(x, y) of the Stokes parameters isincident on an imaging polarimeter using a birefringent prism pair. Acomplex representation of each vibration component is obtained accordingto Expressions (1.17a) to (1.17d) as follows.F ₀ ⁽⁰⁾(x, y)=K ₀(x, y)S ₀ ⁽⁰⁾(x, y)   (1.28a)F ⁻ ⁽⁰⁾(x, y)=K ⁻(x, y)S ₂₃ ⁽⁰⁾(x, y)   (1.28b)F ₂ ⁽⁰⁾(x, y)=K ₂(x, y)S ₁ ⁽⁰⁾(x, y)   (1.28c)F ₊ ⁽⁰⁾(x, y)=K ₊(x, y)S ₂₃ ^((0)*)(x, y)   (1.28d)

Here, the complex functions K₀(x, y), K⁻(x, y), K₂(x, y), and K₊(x, y)are in amount (reference complex function) determined based not uponlight under measurement, but only upon the reference amplitude functionand the reference phase function, as seen from Expressions (1.18a) to(1.18d). Accordingly, these can be calculated backward as follows.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 18} \right\rbrack & \quad \\{{K_{0}\left( {x,y} \right)} = \frac{F_{0}^{(0)}\left( {x,y} \right)}{S_{0}^{(0)}\left( {x,y} \right)}} & \left( {1.29a} \right) \\{{K_{-}\left( {x,y} \right)} = \frac{F_{-}^{(0)}\left( {x,y} \right)}{S_{23}^{(0)}\left( {x,y} \right)}} & \left( {1.29b} \right) \\{{K_{2}\left( {x,y} \right)} = \frac{F_{2}^{(0)}\left( {x,y} \right)}{S_{1}^{(0)}\left( {x,y} \right)}} & \left( {1.29c} \right) \\{{K_{+}\left( {x,y} \right)} = \frac{F_{+}^{(0)}\left( {x,y} \right)}{S_{23}^{{(0)}*}\left( {x,y} \right)}} & \left( {1.29d} \right)\end{matrix}$

Similar to the case of calculating the amplitude and the phaseseparately, once the above reference complex function is obtained (canbe calibrated), a two-dimensional spatial distribution of the Stokesparameters of light having an unknown SOP can be obtained.

It is to be noted that, just for reference, a mathematical expressionsin the case of using linearly polarized light with the azimuth angle 0are shown below. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 19} \right\rbrack & \quad \\{{K_{0}\left( {x,y} \right)} = \frac{F_{0}^{(0)}\left( {x,y} \right)}{I^{(0)}\left( {x,y} \right)}} & \left( {1.30a} \right) \\{{K_{-}\left( {x,y} \right)} = \frac{F_{-}^{(0)}\left( {x,y} \right)}{{I^{(0)}\left( {x,y} \right)}\sin\quad 2\theta}} & \left( {1.30b} \right) \\{{K_{2}\left( {x,y} \right)} = \frac{F_{2}^{(0)}\left( {x,y} \right)}{{I^{(0)}\left( {x,y} \right)}\cos\quad 2\theta}} & \left( {1.30c} \right) \\{{K_{+}\left( {x,y} \right)} = \frac{F_{+}^{(0)}\left( {x,y} \right)}{{I^{(0)}\left( {x,y} \right)}\sin\quad 2\theta}} & \left( {1.30d} \right)\end{matrix}$Chapter 2: Problems of Imaging Polarimeter Using Birefringent Prism Pair

As described in Step 3 in Section 1.4, for demodulation of thetwo-dimensional spatial distributions S₀(x, y), S₁(x, y), S₂(x, y), andS₃(x, y) of Stokes parameters from the measured intensity distributionI(x, y), it is necessary to obtain (calibrate) the following functionsin advance (refer to FIG. 8). $\begin{matrix}{\begin{matrix}{{Reference}\quad} \\{{amplitude}\quad{function}}\end{matrix}\left\{ {\begin{matrix}{m_{0}\left( {x,y} \right)} \\{m_{-}\left( {x,y} \right)} \\{m_{2}\left( {x,y} \right)} \\{m_{+}\left( {x,y} \right)}\end{matrix}\quad\begin{matrix}{Reference} \\{{phase}\quad{function}}\end{matrix}\left\{ {\begin{matrix}{\phi_{1}\left( {x,y} \right)} \\{\phi_{2}\left( {x,y} \right)}\end{matrix}\begin{matrix}{{Or}\quad{reference}\quad{complex}} \\{function}\end{matrix}\left\{ \begin{matrix}{K_{0}\left( {x,y} \right)} \\{K_{-}\left( {x,y} \right)} \\{K_{2}\left( {x,y} \right)} \\{K_{+}\left( {x,y} \right)}\end{matrix} \right.} \right.} \right.} & \left\lbrack {{Mathematical}\quad{Expression}\quad 20} \right\rbrack\end{matrix}$

However, the reference phase functions φ₁(x, y) and φ₂(x, y) have theproperty of varying for a variety of reasons. When these functions vary,there occurs a problem in that a large error occurs in measured valuesof the two-dimensional spatial distribution of the Stokes parameters.

2.1 Cause of Variations in Reference Phase Function

2.1.1 Temperature Change

The reference phase functions φ₁(x, y) and φ₂(x, y) are amounts(retardation) determined by the birefringent prism pairs BPP₁ and BPP₂in the imaging polarimeter. This retardation has the property ofchanging sensitively with respect to a temperature. Hence the phase ofthe intensity distribution is displaced due to the temperature change(refer to FIG. 22). This results in occurrence of an error in a measuredvalue due to a temperature rise (refer to FIG. 23). Moreover, a similarchange occurs with respect to pressure change.

2.1.2 Displacement of Relative Position Between Birefringent Prism Pairand Imaging Element

When a relative position between the birefringent prism pair and theimaging element is displaced, a problem that is “equivalent” tofluctuations in the reference phase function occurs. When the relativeposition between them is displaced in a system to which a relay lens andthe like is inserted, a similar effect to an effect in lateraldisplacement of the intensity distribution is produced. This is anequivalent phase displacement (refer to FIG. 24). In particular, in afield of application in which it is necessary to provide the polarimeteron a mobile object such as remote sensing or a robot vision, therelative position between the birefringent prism pair and the imagingelement is likely to be displaced because vibration is inevitable.

2.1.3 Solution Easily Found

For preventing variations in the reference phase function of eachvibration component, stabilizing a cause of the fluctuations isconsidered. However, this is very hard to realize. For example, whennoting the temperature change, the accuracy required for thetwo-dimensional spatial distribution of an ellipticity angle or anazimuth angle in inspecting an optical electronics is to be not morethan about 0.1°, and for satisfying this, it is necessary to keep thetemperature change within about 0.5° C. This requires large equipmentfor temperature stabilization, unfavorably leading to a loss of avariety of advantages (size reduction, non-inclusion of an activeelement, etc.) of the imaging polarimeter using the birefringent prismpair.

Chapter 3: Constitution of Embodiment of the Present Invention

The reference phase functions φ₁(x, y) and ₁₀₀ ₂(x, y) (depending notupon light under measurement but only upon parameters of thepolarimeter) included in the intensity distribution obtained from theimaging element vary by a variety of factors, which becomes a majorcontributor to an error. In consideration of this respect, in thepresent embodiment, the imaging polarimeter using the birefringent prismpair is provided with a function capable of calibrating the referencephase functions φ₁(x, y) and φ₂(x, y) of each vibration component duringmeasurement (concurrently with measurement) (refer to FIGS. 9 to 11). Inaddition, although only x-section is shown in FIG. 11 for convenience,the same is true in the y-direction.

3.1 Method for Calibration “During Measurement” (No. 1)

The calibration method described in Section 1.5 was a method forcalibration “prior to measurement”. As opposed to this, in the followingsection, a method for calibration “during measurement” is shown. This isan embodiment of the “principal part of the present invention”.

3.1.1 Basic Idea

The amplitude and the phase obtained in Step 2 in Chapter 1 duringmeasurement (when light in an unknown SOP is incident on the imagingpolarimeter using the birefringent prism pair) is shown again below.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 21} \right\rbrack & \quad \\{{{{component}\quad\lbrack 1\rbrack}\quad\left( {{low}\quad{frequency}\quad{component}} \right)}\begin{matrix}{``{amplitude}"} & {{a_{0}\left( {x,y} \right)} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}}} \\{``{phase}"} & {{\delta_{0}\left( {x,y} \right)} = 0}\end{matrix}} & \left( {3.1a} \right) \\{\left. {{{component}\quad\lbrack 2\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{- x}},U_{- y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{-}\left( {x,y} \right)} = {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}}} \\{``{phase}"} & {{\delta_{-}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} - {\phi_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}}}\end{matrix}} & \left( {3.1b} \right) \\{\left. {{{component}\quad\lbrack 3\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{2x}},U_{2y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{2}\left( {x,y} \right)} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}}} \\{``{phase}"} & {{\delta_{2}\left( {x,y} \right)} = {\phi_{2}\left( {x,y} \right)}}\end{matrix}} & \left( {3.1c} \right) \\{\left. {{{component}\quad\lbrack 4\rbrack}\quad\left( {{{central}\quad{spatial}\quad{frequency}\quad U_{+ x}},U_{+ y}} \right)} \right)\begin{matrix}{``{amplitude}"} & {{a_{+}\left( {x,y} \right)} = {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}}} \\{``{phase}"} & {{\delta_{+}\left( {x,y} \right)} = {{\phi_{2}\left( {x,y} \right)} + {\phi_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}} + \pi}}\end{matrix}} & \left( {3.1d} \right)\end{matrix}$

Here, all needed for obtaining the two-dimensional spatial distributionof the four Stokes parameters are found to be:

“amplitude” of component [1]→S₀(x, y)

“amplitude” and “phase” of one of component [2] and component [4]→S₂(x,y) and S₃(x, y)

“amplitude” of component [3]→S₁(x, y)

It is found that the remaining ones as follows are not used fordemodulation of the two-dimensional spatial distribution of the Stokesparameters.

37 phase” of component [3]

“amplitude” and “phase” of the remaining one of components [2] and [4]

The present inventors and the like found it possible to obtain not onlythe two-dimensional spatial distribution of the four Stokes parametersbut also the “reference phase functions (φ₁(x, y) and φ₂(x, y), etc.)”all at once through the use of the remaining component. This methodmeans that calibration can be concurrently performed in the midst ofmeasurement without particular input of known polarized light.

3.1.2 Preparation

In order to use the “calibration method during measurement”, thefollowing prior preparation is necessary.

The reference amplitude functions m₀(x, y), m⁻(x, y), m₂(x, y), andm₊(x, y) are subjected to pre-calibration (refer to FIG. 8)

Since the following method is effective only on the reference phasefunction, any one of the methods described in Section 1.5 is to beperformed as for the reference amplitude function. It is to be notedthat the fluctuations in the reference amplitude function duringmeasurement typically have considerably small magnitude, and areignorable in many cases. Namely, in contrast to the reference phasefunction, there is generally almost no need for re-calibration of thereference amplitude function during measurement.

As for the reference phase function, the pre-calibration is notnecessarily required. However, a relation between φ₁(x, y) and φ₂(x, y)need to be obtained so that one of φ₁(x, y) and φ₂(x, y) can be obtainedfrom the other.

A concrete example to provide the relation between φ₁(x, y) and φ₂(x, y)is shown hereinafter.

EXAMPLE 1

It is assumed that the birefringent prism pairs BPP₁ and BPP₂ are madeof the same medium, and non-linear terms φ₁(x, y) and φ₂(x, y) arenegligible.

At this time, a ratio among coefficients U_(1x), U_(1y), U_(2x), andU_(2y) to determine φ₁(x, y) and φ₂(x, y) can be determined by a ratioof inclination angles of the contact surface. Therefore, when φ₂(x, y)is found, the U_(2x) and U_(2y) can be determined and the U_(1x) andU_(1y) can be obtained from a proportional calculation.

EXAMPLE 2

Similar to the example 1, it is assumed that the birefringent prismpairs BPP₁ and BPP₂ are made of the same medium, and non-linear termsφ₁(x, y) and φ₂(x, y) are negligible.

In this case, when the reference phase function is calibrated inadvance, the ratio among U_(1x), U_(1y), U_(2x), and U_(2y) can bedetermined.

Note here that, in case that φ₁(x, y) and φ₂(x, y) are not negligible,the following “local coefficient” of each pixel may be used instead ofU_(1x), U_(1y), U_(2x), and U_(2y). $\begin{matrix}{{\frac{1}{2\pi}\frac{\partial\phi_{1}}{\partial x}},{\frac{1}{2\pi}\frac{\partial\phi_{1}}{\partial y}},{\frac{1}{2\pi}\frac{\partial\phi_{2}}{\partial x}},{\frac{1}{2\pi}\frac{\partial\phi_{2}}{\partial y}}} & \left\lbrack {{Mathematical}\quad{Expression}\quad 22} \right\rbrack\end{matrix}$In addition, in case that the above ratio differs from pixel to pixel,that is, in case that there is a variation in temperature in a measuringrange, for example, the ratio of the “local coefficient” of each pixelmay be used.

In case that the ratio among U_(1x), U_(1y), U_(2x), and U_(2y) differsduring the measurement (in case that temperatures of the twobirefringent prism pairs are different from each other, for example),the following method cannot be used.

3.1.3 Actual Calibration Method

Based upon the above-mentioned idea, a method for actual calibration isdescribed below.

A. Method for Obtaining Reference Phase Function δ₂(x, v) from VibrationComponent [3]

By noting only vibration component [3], the amplitude and the phasethereof are shown again as follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 23} \right\rbrack & \quad \\\left\{ \begin{matrix}{``{amplitude}"} & {{a_{2}\left( {x,y} \right)} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}}} \\{``{phase}"} & {{\delta_{2}\left( {x,y} \right)} = {\phi_{2}\left( {x,y} \right)}}\end{matrix} \right. & (3.2)\end{matrix}$What needs to be noted here is that the phase δ₂(x, y) of this componentis one of the reference phase functions φ₂(x, y) (itself). Namely, whenthe phase δ₂(x, y) of component [3] is measured, one of the referencephase functions δ₂(x, y) is immediately determined according to thefollowing expression.φ₂(x, y)=δ₂(x, y)   (3.3)

This relational expression is constantly satisfied regardless of an SOPof the light under measurement, meaning that one of the reference phasefunctions can be immediately obtained from a measured value, even fromany kind of light under measurement. This is a calibration method thatcan be performed utterly concurrently during measurement, and in thecase of “using known polarized light”, there is no need for performingcalibration “prior to measurement or after discontinuation ofmeasurement” as in (Section 1.5). However, it should be noted that, atthis time, the condition of observing component [3] at a sufficient SNratio needs to be satisfied (refer to later-described C)

It is to be noted that, when the “complex representation” is obtained inplace of the “set of the amplitude and phase” in Step 2 of the “processfor demodulating two-dimensional spatial distribution of Stokesparameters in Section 1.4, a calculation method, rewritten from theabove and described below, may be applied.

From Expression (1.14b), δ₂(x, y) has the following relation with thecomplex representation F₂(x, y) of component [3].δ₂(x, y)=arg[F ₂(x, y)]  (3.4)

Therefore, the reference phase function phase φ₂(x, y) can be obtainedfrom the complex representation of component [3] according to thefollowing expression.φ₂(x, y)=arg[F ₂(x, y)]  (3.5)It should be noted that what is needed at the time of complexrepresentation is not the reference phase function φ₂(x, y) but thereference complex function K₂(x, y). Since there is a relation betweenthese two functions as expressed by Expression (1.18c), once φ₂(x, y) isdetermined, K₂(x, y) can also be determined (this will be laterdescribed in detail in F).B. Method for Obtaining Reference Phase Function φ₂(x, y) from aPlurality of Vibration Components (Set of [2] and [4], etc.)

The phases of vibration components [2] and [4] are again shown asfollows.

Phase of component [2]:δ⁻(x, y)=φ₂(x, y)−φ₁(x, y)+arg{S ₂₃(x, y)}  (3.6a)

Phase of component [4]:δ₊(x, y)=φ₂(x, y)+φ₁(x, y)−arg{S ₂₃(x, y)}+π  (3.6b)When the one phase is added to the other, φ₁(x, y) and arg {S₂₃(x y)}are canceled out, and only the terms depending upon φ₂(x, y) are left.It is found therefrom that the following expression can be satisfied.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 24} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {{\frac{1}{2}\left\{ {{\delta_{-}\left( {x,y} \right)} + {\delta_{+}\left( {x,y} \right)}} \right\}} - \frac{\pi}{2}}} & (3.7)\end{matrix}$

The right side of the above expression means that one φ₂(x, y) of thereference phase functions can be obtained by taking an average of thephases of vibration components [2] and [4]. Similarly to method A, thisrelational expression can also be satisfied regardless of an SOP oflight under measurement, meaning that one of the reference phasefunctions can be immediately obtained from a measured value, even froman intensity distribution by any kind of light under measurement.

Namely, similar to method A, this is a “calibration method that can beperformed utterly concurrently during measurement”, and in the case of“using known polarized light”, there is no need for performingcalibration “prior to measurement or after discontinuation ofmeasurement” as in (Section 1.5). However, it should be noted that thecondition of observing components [2] and [4] at a sufficient SN rationeeds to be satisfied (refer to later-described C).

Here, similar to the case of method A, a calculation method is describedfor a case where the “complex representation” is obtained in place ofthe “set of the amplitude and phase” in Step 2 of Section 1.4.

From Expression (1.14b), δ⁻(x, y) and δ₊(x, y) have the followingrelation with the complex representations F⁻(x, y) and F₊(x, y) ofcomponents [2] and [4].δ⁻(x, y)=arg [F ⁻(x, y)]  (3.8a)δ₊(x, y)=arg [F ₊(x, y)]  (3.8b)

Therefore, the reference phase function φ₂(x, y) can be obtained fromthe complex representations of the two components as follows.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 25} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {{\frac{1}{2}\left\{ {{\arg\left\lbrack {F_{-}\left( {x,y} \right)} \right\rbrack} + {\arg\left\lbrack {F_{+}\left( {x,y} \right)} \right\rbrack}} \right\}} - \frac{\pi}{2}}} & (3.9)\end{matrix}$Or, the following expression obtained by rewriting the above expressionusing a simple formula of the complex function may be applied.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 26} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {\frac{1}{2}{\arg\left\lbrack {{- {F_{-}\left( {x,y} \right)}}{F_{+}\left( {x,y} \right)}} \right\rbrack}}} & (3.10)\end{matrix}$

In the optical system (the imaging polarimeter using the birefringentprism pairs) in FIG. 1, an obtained intensity distribution containsanother component having a different period as described in FIG. 4 andthe like except for the case where the angle formed between thebirefringent prism pair BPP₂ and the analyzer A is not 45°.

As seen from Expression (1. 9), the phase of this component is “δ₁(x,y)=φ₁(x, y)−arg {S₂₃(x, y)}”, and similar to the phase terms of abovevibration components [2] and [4]. Hence, even when component [2] or [4]is combined (or replaced) with the another component, it is possible tocalibrate φ₂(x, y).

C. Combination of A and B

The two methods (method A and method B) described above are methods inwhich one φ₂(x, y) of the reference phase functions can be calibratedutterly concurrently during measurement. However, the used vibrationcomponents are different between the two methods. What should beconcerned here is that the amplitude of vibration component [3] used inmethod A is proportional to S₁(x, y), while the amplitudes of vibrationcomponents [2] and [4] used in Method B are proportional to thefollowing.|S ₂₃(x, y)|=√{square root over (S ₂ ²(x, y)+S ₃ ²(x,y))}  [Mathematical Expression27]

Since an SOP of light under measurement is unknown, there is noguarantee that the two-dimensional spatial distribution of Stokesparameters is constantly sufficiently large for phase measurement foreach component. For example, when light with small S₁(x, y) is projectedas the light under measurement, determination of φ₂(x, y) by Method Ausing the phase of this component might result in occurrence of a largeerror. For solving this problem, adaptive combination of methods A and Bis desired. Specifically, a value of one φ₂(x, y) with more certaintycan be obtained by selecting, or weighting up and balancing results ofthe two methods.

It should be noted that the light under measurement whose S₁(x, y) andS₂₃(x, y) are “both” very small is practically non-existent. This isbecause, when both are small, an intensity of a complete polarizedlight:√{square root over (S ₁ ²(x, y)+S ₂ ²(x, y)+S ₃ ²(x, y))}  [MathematicalExpression 28]is small, namely light is in a state infinitely close to non-polarizedlight. In such a case, there is no point of obtaining an SOP itself.Accordingly, the combination of the above methods A and B enablescalibration of φ₂(x, y) of light under measurement in any SOPconcurrently with measurement.D. Combination of A and B (No. 2)

One idea for efficiently combining A and B is shown below. This is amethod in which direct calculation is possible without particularseparation by case. It should be noted that, in this part (method D),three complex representation functions F⁻(x, y), F₂(x, y) and F₊(x, y)of vibration components [2] to [4] are used for calculation. When avibration calculation is to be made from the “set of the amplitude andphase” of each vibration component, the set may once be changed to the“complex representation” according to Expression (1.13), and then thefollowing calculation process may be performed.

As a preparation for explaining this method, first, the following twoexpressions are derived and the natures thereof are described. Bytransforming Expression (3.5), the following expression can be obtained.2φ₂(x, y)=arg[F ₂ ²(x, y)]  (3.11)Meanwhile, by doubling both sides of Expression (3.10), the followingexpression can be obtained.2φ₂(x, y)=arg[−F ⁻(x, y)F ₊(x, y)]  (3.12)It is found from the comparison between the above two expressions thatthe complex function in the brackets on the right side of each of theexpressions has the same argument 2φ₂(x, y). Further, when the absolutevalue of the complex function in the brackets is calculated in each ofthe expressions, the results are found to be as follows: $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 29} \right\rbrack & \quad \\{{{F_{2}^{2}\left( {x,y} \right)}} = {\frac{1}{16}{m_{2}^{2}\left( {x,y} \right)}{S_{1}^{2}\left( {x,y} \right)}}} & \left( {3.13a} \right) \\{{{{- {F_{-}\left( {x,y} \right)}}{F_{+}\left( {x,y} \right)}}} = {\frac{1}{64}{m_{-}\left( {x,y} \right)}{m_{+}\left( {x,y} \right)}\left\{ {{S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}} \right\}}} & \left( {3.13b} \right)\end{matrix}$

This expression means that the absolute value of the former (obtainedfrom component [3]) is proportional to S₁ ²(x, y), while the absolutevalue of the latter (obtained from components [2] and [4]) isproportional to S₂ ²(x, y)+S₃ ²(x, y). As described above, these twocomplex functions do not concurrently become smaller. Thereby,appropriate “weighting functions α(x, y) and β(x, y) having the sameargument” were respectively multiplied by the above two complexfunctions, and then the obtained two terms were added together.

[Mathematical Expression 30]α(x, y)[F ₂ ²(x, y)]+β(x, y) [−F ⁻(x, y)F ₊(x, y)]  (3.14)It is revealed that the absolute value of the sum of the two terms(practically) do not become smaller. When either one of S₁ ²(x, y) andS₂ ²(x, y) +S₃ ²(x, y) becomes smaller, one of the above two termsaccordingly becomes smaller, but the other remains for certain. Evenwhen the SOP of the light under measurement changes as a result, theabsolute value of this expression does not become extremely smaller.Further, the argument of this expression is constantly equivalent to2φ₂(x, y)+arg α(x, y). Through the use of these natures, it is possibleto obtain φ₂(x, y) according to the following expression without adecrease in S/N ratio. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 31} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {{\frac{1}{2}\arg\begin{Bmatrix}{{{\alpha\left( {x,y} \right)}\left\lbrack {F_{2}^{2}\left( {x,y} \right)} \right\rbrack} +} \\{{\beta\left( {x,y} \right)}\left\lbrack {{- {F_{-}\left( {x,y} \right)}}{F_{+}\left( {x,y} \right)}} \right\rbrack}\end{Bmatrix}} - {\frac{1}{2}{\arg\left\lbrack {\alpha\left( {x,y} \right)} \right\rbrack}}}} & (3.15)\end{matrix}$

Two ways to select specific α(x, y) and β(x, y) are shown below.

[D-1] α(x, y)=β(x, y)=1

The simplest way to select the weighting functions is making the twofunctions the same constant (1). In this case, an expression forobtaining the reference phase function φ₂(x, y) is shown below.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 32} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {\frac{1}{2}\arg\left\{ {\left\lbrack {F_{2}^{2}\left( {x,y} \right)} \right\rbrack + \left\lbrack {{- {F_{-}\left( {x,y} \right)}}{F_{+}\left( {x,y} \right)}} \right\rbrack} \right\}}} & (3.16)\end{matrix}$ $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 33} \right\rbrack & \quad \\{{\left\lbrack {D - 2} \right\rbrack\quad{\alpha\left( {x,y} \right)}} = {{\frac{16}{m_{2}^{2}\left( {x,y} \right)} \cdot {\beta\left( {x,y} \right)}} = \frac{64}{{m_{-}\left( {x,y} \right)}{m_{+}\left( {x,y} \right)}}}} & \quad\end{matrix}$

Another Example is a method for selecting α(x, y) and β(x, y) usingreference amplitude functions having been subjected to pre-calibration,as shown in the above expression. Here, an expression for deriving thereference phase function φ₂(x, y) from the complex representation of thedemodulated vibration component is shown as below. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 34} \right\rbrack & \quad \\{{\phi_{2}\left( {x,y} \right)} = {\frac{1}{2}\arg\left\{ {\left\lbrack {16\frac{F_{2}^{2}\left( {x,y} \right)}{m_{2}^{2}\left( {x,y} \right)}} \right\rbrack + \left\lbrack {{- 64}\frac{{F_{-}\left( {x,y} \right)}{F_{+}\left( {x,y} \right)}}{{m_{-}\left( {x,y} \right)}{m_{+}\left( {x,y} \right)}}} \right\rbrack} \right\}}} & (3.17)\end{matrix}$With the expression made in this form, $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 35} \right\rbrack & \quad \\{{{\{\}}\quad{The}\quad{absolute}\quad{value}\quad{in}\quad{\{\}}\quad{is}\text{:}}{{{\left\lbrack {16\frac{F_{2}^{2}\left( {x,y} \right)}{m_{2}^{2}\left( {x,y} \right)}} \right\rbrack + \left\lbrack {{- 64}\frac{{F_{-}\left( {x,y} \right)}{F_{+}\left( {x,y} \right)}}{{m_{-}\left( {x,y} \right)}{m_{+}\left( {x,y} \right)}}} \right\rbrack}} = {{S_{1}^{2}\left( {x,y} \right)} + {S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{3}\left( {x,y} \right)}}}\sqrt{{S_{1}^{2}\left( {x,y} \right)} + {S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}}} & (3.18)\end{matrix}$

This absolute value is a square of the following intensity of completepolarized light component of the light under measurement.

In particular, this constantly agrees with the square S₀ ²(x, y) ofintensity of the light under measurement (regardless of the SOP).Namely, Ω₂(x, y) can be constantly stably obtained using Expression(3.17) so long as the light under measurement has sufficient lightintensity.

E. Calculation of φ₁(x, y)

Since fluctuations in φ₁(x, y) are considered to be similar to those inφ₂(x, y), it is possible to obtain φ₁(x, y) by proportional calculation(e.g. by using a relation between the φ₂(x, y) and φ₁(x, y)) from ameasured value of φ₂(x, y).

F. Calculation of Reference Complex Function

In the demodulation in Step 2 of the “process for demodulatingtwo-dimensional spatial distribution of Stokes parameters” in Section1.4, when (not the “set of the amplitude and phase” but) the “complexrepresentation” is obtained, what are needed ultimately in the operationof Step 3 for obtaining the two-dimensional spatial distribution of theStokes parameters are not the reference phase functions φ₁(x, y) andφ₂(x, y) but the reference complex functions K₀(x, y), K⁻( x, y), K₂(x,y), and K₊(x, y). However, these can also be immediately obtainedthrough the use of the relations of Expressions (1.18a) to (1.18d) ifthe reference phase functions φ₁(x, y) and φ₂(x, y) have been obtainedby the processes up to above Process E.

The imaging polarimetry described in this section can be summarized asfollows. In any case, it is assumed that data showing a relation betweenφ₁(x, y) and φ₂(x, y) has been made available.

The imaging polarimetry of this section is a method in which, by the useof an intensity distribution obtained by launching the light undermeasurement into the optical system (polarimetric imaging device) of theimaging polarimeter using the birefringent prism pair, components [1]and [3] of the intensity distribution are obtained, and at least one ofcomponents [2], [4] and [5] is obtained, and by the use of the datashowing the relation between φ₁(x, y) and φ₂(x, y) and each of theobtained intensity distribution components, φ₁(x, y) and φ₂(x, y) areobtained, and also a parameter indicating the two-dimensional spatialdistribution of the SOP is obtained.

More specifically, the imaging polarimetry of method A of this sectionis a method to be performed as follows. By the use of the intensitydistribution obtained by launching the light under measurement into thepolarimetric imaging device, components [1] and [3] of the intensitydistribution are obtained, and at least one of components [2], [4] and[5] is obtained. φ₂(x, y) is obtained from obtained component [3], andφ₁(x, y) is obtained from the data showing the relation between φ₁(x, y)and φ₂(x, y) and obtained φ₂(x, y). By the use of each obtainedintensity distribution component, and obtained φ₁(x, y) and φ₂(x, y), aparameter indicating the two-dimensional spatial distribution of the SOPis obtained method A is a preferred embodiment when the two-dimensionalspatial distribution S₁(x, y) of the Stokes parameter of the light undermeasurement is not zero or not close to zero.

The imaging polarimetry of method B of this section is a method to beperformed as follows. By the use of the intensity distribution obtainedby launching the light under measurement into the polarimetric imagingdevice, components [1] and [3] of the intensity distribution areobtained, and at least two of components [2], [4] and [5] are obtained.φ₂(x, y) is obtained from at least two of components [2], [4] and [5],and φ₁(x, y) is obtained from the data showing the relation betweenφ₁(x, y) and φ₂(x, y) and obtained φ₂(x, y). By the use of each obtainedintensity distribution component, and obtained φ₁(x, y) and φ₂(x, y), aparameter indicating the two-dimensional spatial distribution of the SOPis obtained. The method B is a preferred embodiment in a case other thanthe case where the both two-dimensional spatial distributions S₂(x, y)and S₃(x, y) of the Stokes parameters of the light under measurement areneither zero nor close to zero.

The imaging polarimetry of methods C and D of this section are methodsto be performed as follows. By the use of the intensity distributionobtained by launching the light under measurement into the polarimetricimaging device, components [1] and [3] of the intensity distribution areobtained, and at least two of components [2], [4] and [5] are obtained.φ₂(x, y) is obtained by selecting either a first process for obtainingφ₂(x, y) from the obtained component [3] or a second process forobtaining φ₂(x, y) from at least two of components [2], [4] and [5], orby combining the first process and the second process, and φ₁(x, y) isobtained from the data showing the relation between φ₁(x, y) and φ₂(x,y) and the obtained φ₂(x, y). By the use of each obtained intensitydistribution component, and the obtained φ₁(x, y) and φ₂(x, y), aparameter indicating two-dimensional spatial distribution of the SOP isobtained. The methods C and D are embodiments capable of measurement byappropriate selection of either the first or second process or byappropriate combination of the first and second processes, so long asall of the two-dimensional spatial distribution S₁(x, y), S₂(x, y) andS₃(x, y) of Stokes parameters of the light under measurement are notconcurrently zero or close to zero.

In the imaging polarimetry of this section, since component [5] of theintensity distribution does not appear when the analyzer A is arrangedsuch that the direction of the transmission axis thereof forms an angleof 45° with respect to the direction of the principal axis of the secondbirefringent prism pair BPP₂, at least either or both of components [2]and [4] may be obtained in a part where at least one or two ofcomponents [2], [4] and [5] are to be obtained.

3.2 Method for Calibrating Reference Phase Function “During Measurement”(No. 2)

3.2.1 Basic Idea

In the same idea as described in the previous section 3.1, “only adifference” of the reference phase functions can be obtained. Althoughthe terms “pre-calibration” and “initial value” are used below for thesake of convenience, the timing for calibration is not necessarily priorto measurement of the light under measurement. Therefore, the initialvalue of the reference phase function is typically grasped as areference value for calibration of the reference phase function.Further, an appropriate value which is not a measured value is usable asthe reference value for calibration of the reference phase function.

In the previous method (in the previous section 3.1), the “referenceamplitude function” was obtained in the pre-calibration, and it was notparticularly necessary to obtain the “reference phase function”.However, as appeared from Section 3.2, those two functions can becalibrated almost concurrently. It is thus possible to obtain in advancean “initial value of the reference phase function in pre-calibration” soas to only track a difference thereof during measurement.

Advantages in this case are as follows.

Slightly additional phase displacement part which might be generated dueto properties of the imaging element or the signal processing system canbe removed.

Burdensome phase unwrapping is not necessary.

Since a phase difference itself is small, a dynamic range in calculationcan be made small. Further, as a result of this, a calculation error canbe relatively made small in many cases.

Accordingly, “obtaining only the difference in the reference phasefunction” has its own meaning.

The following is described as supplement for the foregoing explanation.As shown in FIG. 11, the two methods have different factors of an errorin calculation of φ₁ from φ₂ Namely, as shown in FIG. 11(a), it isnecessary to perform phase unwrapping for obtaining φ₁(x, y) from φ₂(x,y). This phase unwrapping is a major factor of the error. Especiallywhen period frequency is high as compared with sampling, noise isincluded in the period, or the like, wrong phase unwrapping might beperformed. With wrong phase wrapping performed, an error becomes aninteger multiple of 2π, leading to calculation of a wrong phase.Further, this error affects a broad region. The error is essentiallycaused by that a solution of an arg operator (or an arctan operator) forobtaining an argument has phase ambiguity by the integer multiple of 2π.As opposed to this, as shown in FIG. 11(b), it is not necessary inobtaining Δφ₁(x, y) from Δφ₂(x, y) to perform phase unwrapping since thedifference Δφ₂(x, y) from the initial value of the reference phasefunction is small. This allows the measurement error to be relativelysmall.

3.2.2 Preparation

The use of the “calibration method during measurement” is based upon thepremise of pre-calibration of both the “reference amplitude function”and the “reference phase function” prior to measurement. It is to benoted that, as for the phase, an obtained value of the phase is notnecessarily required to have high accuracy since a variance difference,i.e. an error, can be corrected later.

In addition, it is necessary to find the relation between the“difference of the reference phase function” Δφ₁(x, y) and Δφ₂(x, y) inadvance. In this case, the following examples are provided.

EXAMPLE 1

The relation between the φ₁(x, y) and φ₂(x, y) is used as it is.

EXAMPLE 2

The relation between the Δφ₁(x, y) and Δφ₂(x, y) is found by applyingfluctuation (temperature change, for example) actually.

3.2.3 Actual Calibration Method

The basic idea on the calibration method is completely the same as inSection 3.1. There thus exist calculation methods corresponding to all Ato E described in Section 3.1.3. Hence, in this section, the idea isdescribed only when different from that of the previous section, and thefollowing description concentrates on listing of mathematicalexpressions.

First, a couple of symbols are defined. The reference phase functionsobtained by the pre-calibration are defined as φ₁ ^((i))(x, y) and φ₂^((i))( x, y). Reference complex functions corresponding to thesereference phase functions are expressed as follows according toExpressions (1.18a) to (1.18d). $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 36} \right\rbrack & \quad \\{{K_{0}^{(i)}\left( {x,y} \right)} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}}} & \left( {3.19a} \right) \\{{K_{-}^{(i)}\left( {x,y} \right)} = {\frac{1}{8}{m_{-}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\left\lbrack {{\phi_{2}^{(i)}\left( {x,y} \right)} - {\phi_{1}^{(i)}\left( {x,y} \right)}} \right\rbrack}}} & \left( {3.19b} \right) \\{{K_{2}^{(i)}\left( {x,y} \right)} = {\frac{1}{4}{m_{2}\left( {x,y} \right)}\exp\quad{{\mathbb{i}\phi}_{2}^{(i)}\left( {x,y} \right)}}} & \left( {3.19c} \right) \\{{K_{+}^{(i)}\left( {x,y} \right)} = {{- \frac{1}{8}}{m_{+}\left( {x,y} \right)}\exp\quad{{\mathbb{i}}\quad\left\lbrack {{\phi_{2}^{(i)}\left( {x,y} \right)} + {\phi_{1}^{(i)}\left( {x,y} \right)}} \right\rbrack}}} & \left( {3.19d} \right)\end{matrix}$Assuming that the reference phase functions changed during measurementas follows.φ₁(x, y)=φ₁ ^((i))(x, y)+Δφ₁(x, y)   (3.20a)φ₂(x, y)=φ₂ ^((i))(x, y)+Δφ₂(x, y)   (3.20b)

Below described are methods for obtaining the differences Δφ₁(x, y) andΔφ₂(x, y) of the reference phase functions or changes in the referencecomplex functions corresponding to those differences.

A. Method for Obtaining Reference Phase Function φ₂(x, y) from VibrationComponent [3]

As described in method A in the previous section, the phase of component[3] is expressed as follows.δ₂(x, y)=φ₂(x, y)=φ₂ ^((i))(x, y)+δφ₂(x, y)   (3.21)Here, the difference in φ₂(x, y) can be obtained as:Δφ₂(x, y)=δ₂(x, y)−φ₂ ^((i))(x, y)   (3.22)Namely, this means that, once the phase δ₂ of component [3] is measured,one δφ₂(x, y)) of the differences in the reference phase functions canbe immediately determined.

It is to be noted that in Step 2, when not the “set of amplitude andphase” but the “complex representation” is obtained, it is obtainedaccording to the following expressions. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 37} \right\rbrack & \quad \\{{\delta_{2}\left( {x,y} \right)} = {\arg\left\lbrack {F_{2}\left( {x,y} \right)} \right\rbrack}} & \left( {3.23a} \right) \\{{{\phi_{2}^{(i)}\left( {x,y} \right)} = {\arg\left\lbrack {K_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}}{From}} & \left( {3.23b} \right) \\{{{\Delta\quad{\phi_{2}\left( {x,y} \right)}} = {{\arg\left\lbrack {F_{2}\left( {x,y} \right)} \right\rbrack} - {\arg\left\lbrack {K_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}}}{Or}} & (3.24) \\{{\Delta\quad{\phi_{2}\left( {x,y} \right)}} = {\arg\left\lbrack \frac{F_{2}\left( {x,y} \right)}{K_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}} & (3.25)\end{matrix}$B. Method for Obtaining Reference Phase Function φ₂(x, y) from aPlurality of Vibration Components (set of [2] and [4]. etc.)

In the method for obtaining the difference in φ₂(x, y) from the phase ofeach of vibration component [2] and [4], the difference is obtainedaccording to the following expression. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 38} \right\rbrack & \quad \\{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\left\lbrack {{\frac{1}{2}\left\{ {{\delta_{-}\left( {x,y} \right)} + {\delta_{+}\left( {x,y} \right)}} \right\}} - \frac{\pi}{2}} \right\rbrack - {\phi_{2}^{(i)}\left( {x,y} \right)}}} & (3.26)\end{matrix}$

When not the “set of amplitude and phase” but the “complexrepresentation” is to be obtained, the difference is obtained accordingto the following expressions. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 39} \right\rbrack & \quad \\{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}\left\{ {{\arg\left\lbrack {F_{-}\left( {x,y} \right)} \right\rbrack} + {\arg\left\lbrack {F_{+}\left( {x,y} \right)} \right\rbrack} - {\arg\left\lbrack {K_{-}^{(i)}\left( {x,y} \right)} \right\rbrack} - {\arg\left\lbrack {K_{+}^{(i)}\left( {x,y} \right)} \right\rbrack}} \right\}}} & (3.27)\end{matrix}$Or, the following expressions obtained by rewriting the above expressionusing a simple formula of the complex function may be applied.$\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 40} \right\rbrack & \quad \\{{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}\left\{ {{\arg\left\lbrack \frac{F_{-}\left( {x,y} \right)}{K_{-}^{(i)}\left( {x,y} \right)} \right\rbrack} + {\arg\left\lbrack \frac{F_{+}\left( {x,y} \right)}{K_{+}^{(i)}\left( {x,y} \right)} \right\rbrack}} \right\}}}{Or}} & (3.28) \\{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}{\arg\left\lbrack {\frac{F_{-}\left( {x,y} \right)}{K_{-}^{(i)}\left( {x,y} \right)}\frac{F_{+}\left( {x,y} \right)}{K_{+}^{(i)}\left( {x,y} \right)}} \right\rbrack}}} & (3.29)\end{matrix}$In addition, as noted at the end of Section 3.1.3, the same idea asabove shown can be applied to the case of using another term.C. Combination of A and B

As in the case described in the previous section, adaptive combinationof methods A and B is also effective in the case of obtaining only the“difference” in the reference phase functions. It should be noted that adescription of the combination is completely the same as that in theprevious section and it is thus omitted.

D. Combination of A and B (No. 2)

One of desired mathematical expressions in the case of obtaining onlythe difference is as follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 41} \right\rbrack & \quad \\{{\alpha\left( {x,y} \right)} = \left\lbrack \frac{1}{K_{2}^{(i)}\left( {x,y} \right)} \right\rbrack^{2}} & \left( {3.30a} \right) \\{{\beta\left( {x,y} \right)} = {- \frac{1}{{K_{-}^{(i)}\left( {x,y} \right)}{K_{+}^{(i)}\left( {x,y} \right)}}}} & \left( {3.30b} \right)\end{matrix}$Since arg[α(x, y)]=arg[β(x, y)]=2φ₂(x, y) in the above expressions, thedifference can be obtained as follows. $\begin{matrix}\left\lbrack {{Mathematcial}\quad{Expression}\quad 42} \right\rbrack & \quad \\{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}\arg\left\{ {\left\lbrack \frac{F_{2}\left( {x,y} \right)}{K_{2}^{(i)}\left( {x,y} \right)} \right\rbrack^{2} + {\frac{F_{-}\left( {x,y} \right)}{K_{-}^{(i)}\left( {x,y} \right)}\frac{F_{+}\left( {x,y} \right)}{K_{+}^{(i)}\left( {x,y} \right)}}} \right\}}} & (3.31)\end{matrix}$This absolute value is a square of the following intensity of completepolarized light component of light under measurement:√{square root over (S ₁ ²(x, y)+S ₂ ²(x, y)+S₃ ²(x, y))}In particular, this is constantly the square S₀ ²(x, y) of intensity ofthe light under measurement (regardless of the SOP) in the case ofcomplete polarized light. Namely, Δφ₂(x, y) can be constantly stablyobtained using the above expression so long as the light undermeasurement has sufficient light intensity.E. Calculation of Δφ₁(x, y)

Fluctuations in Δφ₁(x, y) are considered to be similar to those inΔφ₂(x, y). They can be obtained from the measured value of the Δφ₂(x, y)using a proportional calculation (the relation between Δφ₂(x, y) andΔφ₁(x, y)).

F. Calculation of Reference Complex Function

In the demodulation of each vibration component in Step 2, when not the“set of the amplitude and phase” but the “complex representation” isobtained, what are needed ultimately in obtaining the two-dimensionalspatial distribution of the Stokes parameters (operation of Step 3) arenot the reference phase functions φ₁(x, y) and φ₂(x, y) but thereference complex functions K₀(x, y), K⁻( x, y), K₂(x, y), and K₊(x, y).

If the reference phase function differences Δφ₁(x, y) and Δφ₂(x, y) havebeen obtained by the processes up to above Process E, the referencecomplex functions can be immediately obtained as follows.

[Mathematical Expression 43]K ₀(x, y)=K ₀ ^((i))(x, y)   (3.32a)K ⁻(x, y)=K ⁻ ^((i))(x, y)e ^(i{Δφis 2) ^((x,y)−Δφ) ¹ ^((x,y)})  (3.32a)K ₂(x, y)=K ₂ ^((i))(x, y)e ^(iΔφ) ¹ ^((x,y)})  (3.32c)K ₊(x, y)=K ₊ ^((i))(x, y)e ^(i{Δφ) ² ^((x,y)+Δφ) ¹ ^((x,y)})  (3.32d)

The imaging polarimetry described in this section can be summarized asfollows. In any case, it is assumed that a reference value forcalibration of the first reference phase function φ₁ ^((i))(x, y), areference value for calibration of the second reference phase functionφ₂ ^((i))(x, y), and data showing the relation between Δφ₁(x, y) andΔφ₂(x, y) are made available.

The imaging polarimetry of this section is a method to be performed asfollows. By the use of the intensity distribution obtained by launchingthe light under measurement into the polarimetric imaging device,components [1] and [3] of the intensity distribution are obtained, andat least one of components [2], [4] and [5] is obtained, and by the useof φ₁ ^((i))(x, y), φ₂ ^((i))(x, y), the data showing the relationbetween Δφ₁(x, y) and Δφ₂(x, y), and each of the obtained intensitydistribution components, Δφ₁(x, y) and Δφ₂(x, y) are obtained, and alsoa parameter indicating two-dimensional spatial distribution of the SOPis obtained.

More specifically, the imaging polarimetry of method A of this sectionis a method to be performed as follows. By the use of the intensitydistribution obtained by launching the light under measurement into thepolarimetric imaging device, components [1] and [3] of the intensitydistribution are obtained, and at least one of components [2], [4] and[5] is obtained. Δφ₂(x, y) is obtained from the obtained component [3]and Δφ₁(x, y) is obtained from the obtained Δφ₂(x, y). By the use ofeach of the obtained intensity distribution components and Δφ₁(x, y) andΔφ₂(x, y), a parameter indicating two-dimensional spatial distributionof the SOP is obtained.

The imaging polarimetry of method B of this section is a method to beperformed as follows. By the use of the intensity distribution obtainedby launching the light under measurement into the polarimetric imagingdevice, components [1] and [3] of the intensity distribution areobtained, and at least two of components [2], [4] and [5] are obtained.Δφ₂(x, y) is obtained from at least two of components [2], [4] and [5],and Δφ₁(x, y) is obtained from the obtained Δφ₂(x, y). By the use ofeach obtained spectral intensity component, and the obtained Δφ₁(x, y)and Δφ₂(x, y), a parameter indicating two-dimensional spatialdistribution of the SOP is obtained.

The imaging polarimetry of methods C and D of this section are methodsto be performed as follows. By the use of the intensity distributionobtained by launching the light under measurement into the polarimetricimaging device, components [1] and [3] of the intensity distribution areobtained, and at least two of components [2], [4] and [5] are obtained.Δφ₂(x, y) is obtained by selecting either a first process for obtainingΔφ₂(x, y) from obtained component [3] or a second process for obtainingΔφ₂(x, y) from at least two of components [2], [4] and [5], or bycombining the first process and the second process, and Δφ₁(x, y) isobtained from the obtained Δφ₂(x, y). By the use of each obtainedintensity distribution component, and obtained Δφ₁(x, y) and Δφ₂(x, y),a parameter indicating two-dimensional spatial distribution of the SOPis obtained.

In the imaging polarimetry of this section, since component [5] of theintensity distribution does not appear when the analyzer A is arrangedsuch that the direction of the transmission axis thereof forms an angleof 45° with respect to the direction of the principal axis of the secondbirefringent prism pair BPP₂, at least either or both of components [2]and [4] may be obtained in a part where at least one or two ofcomponents [2], [4] and [5] are to be obtained.

Chapter 4: Common Demonstration of Possibility for Calibration DuringMeasurement

As described in the previous chapter, it is possible in the imagingpolarimetry using the birefringent prism pair to calibrate (or correct)a reference phase function or a reference phase function difference“during measurement (concurrently with measurement)”. However, thedescription given in the previous chapter was based upon the premise ofapplying a signal processing method using frequency filtering, namely,separating a quasi-sinusoidal component that vibrates at a frequencydifferent from an intensity distribution obtained from the imagingelement. However, this frequency filtering is in practice not anessential step for realization of “calibration during measurement”. Theinventors and the like found it possible to perform calibration of thereference phase function during measurement even by a differentdemodulation method, namely, a different signal processing method.

In order to demonstrate this, first in this chapter, the reason whycalibration is possible during measurement in the imaging polarimetryusing the birefringent prism pair is described without limiting to the“specific process of the signal processing method”. Further, in the nextchapter, a “method for applying a generalized inverse matrix” is shownas a specific example of a “calibration method during measurementwithout the use of frequency filtering”.

4.1 Relation Between Intensity Distribution Obtained from ImagingElement and Reference Phase Functions Δφ₁(x, y), Δφ₂(x, y)

First of all, the relation between the intensity distribution obtainedfrom the imaging element and the reference phase function is describedusing an idea of interference. In the lower part of FIG. 25, two upperand lower lines traveling in parallel are channels of the two linearlypolarized light components which are orthogonal to each other. However,the respective directions of the linearly polarized lights in thebirefringent prism pairs BPP₁ and BPP₂ are assumed to be arranged alongthe principal axes of the respective elements. Light entered from theleft into the birefringent prism pair BPP₁ is separated into x and ypolarized light components (E_(x)(x, y) and E_(y)(x, y)), and theseparated components propagate respectively along two principal axes ofthe BPP₁ in the directions 0° and 90°. The directions of the principalaxes of the two linearly polarized light components emitted from theBPP₁ are rotated at 45° prior to incidence on the BPP₂, and at thattime, part of the polarized light component is exchanged. The light isredistributed to components along the two principal axes of the BPP₂ inthe directions of 45° and 135°, and transmits through the BPP₂. The twocomponents emitted from the BPP₂ are superposed on each other in theanalyzer A, and then incident on the imaging element.

As immediately apparent by tracing the channels in this figure, thereexist four channels from the incidence end to the imaging element asshown below.

E_(x)(x, y)→principal axis of BPP₁ in 0° direction→principal axis ofBPP₂ in 45° direction→imaging element

E_(x)(x, y)→principal axis of BPP₁ in 0° directions→principal axis ofBPP₂ in 135° direction—imaging element

E_(x)(x, y)→principal axis of BPP₁ in 90° direction—principal axis ofBPP₂ in 45° direction—imaging element

E_(x)(x, y)→principal axis of BPP₁ in 90° direction—principal axis ofBPP₂ in 135° direction—imaging element

In the imaging element, these four components are superposed on oneanother to mutually interfere. A phase of an interference term isdetermined from a phase difference between arbitral two components takenout from the four components. All possible combined sets of componentsare listed below.

-   -   0    -   φ₂(x,y)    -   {φ₁(x,y)−δ(x,y)}    -   φ₂(x,y)−{φ₁(x,y)−δ(x,y)}    -   φ₂(x,y)+{φ₁(x,y)−δ(x,y)}        However, δ (x, y) is a phase difference between the x and y        polarized light components of light under measurement, namely,        δ(x, y)=arg[E _(y)(x, y)]−arg[E _(x)(x, y)]=arg[S ₂₃(x,        y)]  (4.1)        The intensity distribution generated in the imaging element        consequently contains vibration components corresponding to five        kinds of phase differences shown above. (However, as described        in Section 1.2, when the crossing angle between BPP₂ and A is        45°, the terms depending upon {φ₁(x, y)−δ₁(x, y)} are canceled        out, and thus do not occur in the intensity distribution.)

Here, in the combinations of phase differences that appear in theintensity distribution, the way φ₁(x, y) and φ₂(x, y) appear isexamined. φ₁(x, y) constantly appears as a difference from the phasedifference δ₁(x, y)=arg[S₂₃(x, y)] between the x and y polarized lightcomponents of light under measurement, namely {φ₁(x, y)−δ(x, y)}. On theother hand, φ₂(x, y) appears independently, or as the sum with ordifference from {φ₁(x, y)−δ(x, y)}. Thereby, the following is found.

As for φ₁(x, y), when the SOP of light under measurement is unknown, itis not possible to obtain the value directly from the intensitydistribution obtained from the imaging element alone. This is becausethe value can be obtained only as {φ₁(x, y)−δ₁(x, y)}, and φ₁(x, y)cannot be specified when the phase difference δ (x, y) between the x andy polarized light components of the light under measurement is unknown.

On the other hand, as for φ₂(x, y), there is no limitation as in thecase of φ₁(x, y). There is a term independently containing φ₂(x, y).Other terms contain φ₂(x, y) as a sum with or a difference from {φ₁(x,y)}−δ(x, y)}, and therefore an average between the two may be taken.Namely, φ₂(x, y) contained in the intensity distribution obtained fromthe imaging element can be constantly fixed even when the SOP,especially the phase difference δ(x, y) between the x and y polarizedlight components, of the light under measurement takes any value.Namely, this means that calibration concurrently with measurement ispossible as for φ₂(x, y).

It is to be noted that, once the φ₂(x, y) is obtained, the φ₁(x, y) mayalso be indirectly obtained in many cases. This is because there areoften cases where the φ₁(x, y) and the φ₂(x, y) are under the samedisturbance, and also the relation between the φ₁(x, y) and the φ₂(x, y)is known in advance. Namely, Once the φ₂(x, y) is fixed from theintensity distribution obtained from the imaging element, the φ₁(x, y)can be fixed according to a relation between them known in advance.

The basic principle obtained above is summarized as follows.

With appropriate signal processing performed, it is possible todemodulate the φ₂(x, y) from the intensity distribution obtained fromthe imaging element regardless of the SOP of the light undermeasurement, namely without the use of information provided in advanceon the SOP of the light under measurement.

Through the use of the relation between the φ₁(x, y) and the φ₂(x, y),it is possible to also demodulate the φ₁(x, y), though indirectly,independently of the SOP of the light under measurement.

It is to be noted that, what needs to be concerned here is apparentlythat φ₂(x, y) is not necessarily obtained in advance of the φ₂(x, y),depending upon a formula making manner. When the relation between theφ₁(x, y) and the φ₂(x, y) is given in advance and a formula is madeincluding such a relation, it may be represented (at least in amathematical expression,) that the φ₁(x, y) is obtained concurrentlywith the φ₂(x, y) or in advance of the φ₂(x, y).

4.2 Phase Attribute Function of Measurement System

In the previous section, it was demonstrated that the reference phasefunction φ₂(x, y) can be obtained independently of the SOP of the lightunder measurement. Here, this principle does not mean that the φ₂(x, y)itself needs to be directly obtained. The ways to obtain the φ₂(x, y)may for example include obtaining the difference Δφ₂(x, y) from theinitial value φ₂ ^((i))(x, y) when it is known. Or an amount includingthe reference phase function φ₂(x, y) and the like, e.g. K₂(x, y), cosφ₂(x, y), cos Δφ₂(x, y), etc. can be obtained during measurement.

Further, when the relation between the φ₁(x, y) and the φ₂(x, y) isknown in advance, an expression including φ₁(x, y), the differenceΔφ₁(x, y) thereof and the like, e.g. K⁻(x, y), K₊(x, y), cos [φ₂(x,y)−φ₁(x, y)], cos [Δφ₂(x, y)−Δφ₁(x, y)], etc. can all be calibratedduring measurement, and using these, it is possible to concurrentlymeasure the two-dimensional spatial distribution of the Stokesparameters, or polarized light parameters similar to them.

Hereinafter, a function as thus described which is directly orindirectly related to the reference phase functions φ₁(x, y) and φ₂(x,y) and the differences thereof from their reference value, and isdetermined only with parameters of the imaging polarimetric measurementsystem using the birefringent prism pair is referred to as a phaseattribute function of the measurement system. While some of phaseattribute functions are necessary in demodulating the two-dimensionalspatial distribution of the SOP of the light under measurement from theintensity distribution obtained from the imaging element, a function,such as the reference amplitude function, which does not depend upon thereference phase function, may also be necessary in some cases. A set offunctions, which is determined based only upon parameters of the imagingpolarimetry measurement system using the birefringent prism pair and issufficient for demodulation of the two-dimensional spatial distributionof the SOP, is generically named as a set of attribute functions of ameasurement system.

With the use of this term, it can be said that the present invention“provides a method for calibrating, concurrently with measurement ofpolarized light, a set of phase attribute functions out of attributefunctions sufficient for demodulation of a two-dimensional spatialdistribution of an SOP.”

With consideration of the above descriptions, “explicit” frequencyfiltering is found not necessarily essential for obtaining the phaseattribute function of the measurement system from the intensitydistribution provided from the imaging element. Although the operationfor separating some components contained in the intensity distributionis certainly included in the signal processing, the separation is notnecessarily required to be performed with “a period of aquasi-sinusoidal component” on the reference. All needed is separationsufficient for extracting φ₂(x, y) and an amount relative to thedifference of the φ₂(x, y).

Chapter 5: Calibration Method During Measurement Through Use ofGeneralized Inverse Matrix

A method for using a generalized inverse matrix is shown in this chapteras one of specific examples of methods for calibration of a phaseattribute function during measurement and demodulation oftwo-dimensional spatial distribution of Stokes parameters without theuse of frequency filtering, namely without separation of aquasi-sinusoidal vibration component from the intensity distributionobtained from an imaging element.

5.1 Matrix Representation

It is assumed that reference phase functions to be obtained by somepre-calibration are φ₁ ^((i))(x, y)) and φ₂ ^((i))(x, y), and that thereference phase functions are changed as follows during measurement.φ₁(x,y)=φ₁ ^((i))(x,y)+Δφ₁(x,y)   (5.1a)φ₂(x,y)=φ₂ ^((i))(x,y)+Δφ²(x,y)   (5.1b)Below described is a method for obtaining the differences Δφ₁(x, y),Δφ₂(x, y) of the reference phase functions, or a change in referencecomplex functions corresponding to these.

When the above expressions are substituted into Expression (1.5), thefollowing expression is given. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 44} \right\rbrack & \quad \\{{I\left( {x,y} \right)} = {{\frac{1}{2}{m_{0}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)}} + {\frac{1}{4}{m_{-}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}^{(i)}\left( {x,y} \right)} + {{\Delta\phi}_{2}\left( {x,y} \right)} - {\phi_{1}^{(i)}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)} + {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}} + {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {{\phi_{2}^{(i)}\left( {x,y} \right)} + {{\Delta\phi}_{2}\left( {x,y} \right)}} \right\rbrack}} - {\frac{1}{4}{m_{+}\left( {x,y} \right)}{{S_{23}\left( {x,y} \right)}}{\cos\left\lbrack {{\phi_{2}^{(i)}\left( {x,y} \right)} + {{\Delta\phi}_{2}\left( {x,y} \right)} + {\phi_{1}^{(i)}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)} - {\arg\left\{ {S_{23}\left( {x,y} \right)} \right\}}} \right\rbrack}}}} & (5.2)\end{matrix}$Here, when the second and fourth terms (underlined) of this expressionare put together and transformed, the following expression is given.$\begin{matrix}{{I\left( {x,y} \right)} = {{p_{0}\left( {x,y} \right)} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{p_{c}\left( {x,y} \right)}} + {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{p_{s}\left( {x,y} \right)}} + {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\sin\left\lbrack {\phi_{1}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{ss}\left( {x,y} \right)}} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\cos\left\lbrack {\phi_{1}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{cc}\left( {x,y} \right)}} + {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\cos\left\lbrack {\phi_{1}^{(i)}\left( {x,y} \right)} \right\rbrack}q_{sc}{\sin\left\lbrack {\phi_{1}^{(i)}\left( {x,y} \right)} \right\rbrack}} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\sin\left\lbrack {\phi_{1}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{cs}\left( {x,y} \right)}}}} & (5.3)\end{matrix}$However, $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 45} \right\rbrack & \quad \\{{p_{0}\left( {x,y} \right)} = {\frac{1}{2}{m_{0}\left( {x,y} \right)}\quad{S_{0}\left( {x,y} \right)}}} & \left( {5.4a} \right) \\{{p_{c}\left( {x,y} \right)} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\cos\left\lbrack {{\Delta\phi}_{2}\left( {x,y} \right)} \right\rbrack}}} & \left( {5.4b} \right) \\{{p_{s}\left( {x,y} \right)} = {\frac{1}{2}{m_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)}{\sin\left\lbrack {{\Delta\phi}_{2}\left( {x,y} \right)} \right\rbrack}}} & \left( {5.4c} \right) \\{{q_{ss}\left( {x,y} \right)} = {{\frac{1}{4}{m_{-}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} - {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}} + {\frac{1}{4}{m_{+}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} + {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}}}} & \left( {5.4d} \right) \\{{q_{cc}\left( {x,y} \right)} = {{\frac{1}{4}{m_{-}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} - {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}} - {\frac{1}{4}{m_{+}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} + {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}}}} & \left( {5.4e} \right) \\{{q_{sc}\left( {x,y} \right)} = {{{- \frac{1}{4}}{m_{-}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} + {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}} + {\frac{1}{4}{m_{+}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} - {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}}}} & \left( {5.4f} \right) \\{{q_{cs}\left( {x,y} \right)} = {{\frac{1}{4}{m_{-}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} + {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} - {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}} + {\frac{1}{4}{m_{+}\left( {x,y} \right)}\left\{ {{{S_{2}\left( {x,y} \right)}{\cos\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}} - {{S_{3}\left( {x,y} \right)}{\sin\left\lbrack {{{\Delta\phi}_{2}\left( {x,y} \right)} + {{\Delta\phi}_{1}\left( {x,y} \right)}} \right\rbrack}}} \right\}}}} & \left( {5.4g} \right)\end{matrix}$

Incidentally, in the actual measurement, the two-dimensional spatialcoordinates are digitized since digitized measured values are used.Assuming that a digitized marks are denoted as M and N in the x and ydirections and a digitized two-dimensional spatial coordinates aredenoted as x_(m) and y_(n), respectively (m=1 . . . M, n=1 . . . N),Expression (5.3) can be written as follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 46} \right\rbrack & \quad \\{{I\left( {x_{m},y_{n}} \right)} = {{1 \cdot {p_{0}\left( {x_{m},y_{n}} \right)}} + {\left\{ {\cos\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack} \right\} \cdot {p_{c\quad}\left( {x_{m},y_{n}} \right)}} + {\left\{ {\sin\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack} \right\} \cdot {p_{s\quad}\left( {x_{m},y_{n}} \right)}} + {\left\{ {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}{\sin\left\lbrack {\phi_{1}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}} \right\} \cdot {q_{ss}\left( {x_{m},y_{n}} \right)}} + {\left\{ {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}{\cos\left\lbrack {\phi_{1}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}} \right\} \cdot {q_{cc}\left( {x_{m},y_{n}} \right)}} + {\left\{ {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}{\cos\left\lbrack {\phi_{1}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}} \right\} \cdot {q_{sc}\left( {x_{m},y_{n}} \right)}} + {\left\{ {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}{\sin\left\lbrack {\phi_{1}^{(i)}\left( {x_{m},y_{n}} \right)} \right\rbrack}} \right\} \cdot {q_{cs}\left( {x_{m},y_{n}} \right)}}}} & (5.5)\end{matrix}$This expression means that the intensity distribution I(x_(m), y_(n)) isa linear sum of a group of parameters p₀(x_(m), y_(n)), p_(c)(x_(m),y_(n)), p_(s)(x_(m), y_(n)), q_(ss)(x_(m), y_(n), q) _(cc)(x_(m),y_(n)), q_(sc)(x_(m), y_(n)), and q_(cs)(x_(m), y_(n)), which includesthe two-dimensional spatial distribution of the Stokes parameters andthe reference phase function differences. Therefore, this can be writtenin matrix form. Examples of the way for such writing are listed below.P=RQ   (5.6)

Here, as I=(m−1)N+n, elements of a column vector P (line MN), Q (line7MN) in m=1 . . . M, and n=1 . . . N are:P _(I) =I(x _(m) ,y _(n))   (5.7a)Q _((7I−6)) =p ₀(x _(m) ,y _(n))   (5.7b)Q _((7I−5)) =p _(c)(x _(m) ,y _(n))   (5.7c)Q _((7I−4)) =p _(s)(x _(m) ,y _(n))   (5.7d)Q _((7I−3)) =q _(ss)(x _(m) ,y _(n))   (5.7e)Q _((7I−2)) =q _(cc)(x _(m) ,y _(n))   (5.7f)Q _((7I−1)) =q _(sc)(x _(m) ,y _(n))   (5.7g)Q _((7I)) =q _(cs)(x _(m) ,y _(n))   (5.7h)On the other hand, elements of a matrix R (line MN, column 7MN) in m=1 .. . M and n=1 . . . N are:R _(I(7I−6))=1   (5.8a)R _(I(7I−5))=cos[φ₂ ^((i))(x _(m) ,y _(n))]  (5.8b)R _(I(7I−4))=sin[φ₂ ^((i))(x _(m) ,y _(n) )]  (5.8c)R _(I(7I−3))=sin[φ₂ ^((i))(x _(m) ,y _(n))]sin[φ₁ ^((i))(x _(m) ,y_(n))]  (5.8d)R _(I(7I−2))=cos[φ₂ ^((i))(x _(m) ,y _(n))]cos[φ₁ ^((i))(x _(m) ,y_(n))]  (5.8e)R _(I(7I−1))=sin[φ₂ ^((i))(x _(m) ,y _(n))]cos[φ₁ ^((i))(x _(m) ,y_(n))]  (5.8f)R _(I(7I))=cos[φ₂ ^((i))(x _(m) ,y _(n))]sin[φ₁ ^((i))(x _(m) ,y_(n))]  (5.8g)Only the above elements have values and the remaining elements are zero.It is to be noted that in this selection manner, all elements are realnumbers.

Other than the above example, there may exist an almost unlimited numberof ways to represent properties of the imaging polarimeter using thebirefringent prism pair in matrix form. Any representation may beacceptable so long as satisfying the following conditions.

Condition 1: A column vector on the left side (P in the above example)lists information on two-dimensional spatial distribution of lightintensity obtained from the imaging element.

Condition 2: A column vector on the right side (Q in the above example)lists information including two-dimensional spatial distribution ofStokes parameters of light under measurement and phase attributefunctions of a measurement system.

Condition 3: A matrix on the right side (R in the above example) is aliner sum that completely relates the column vectors on the left andright sides to each other, and all elements thereof are fixed beforedemodulation. (A provisional calibration value or the like may be used.)It should be noted that an element of Q made relative to one of elementsof P is not related to other elements of P in the above example.However, this is not essential. If anything, depending upon aconstitution of an optical system, an approximating manner in atheoretical expression, or the like, there may be cases where theabove-mentioned relation between elements of Q and P does not apply,namely, an intensity distribution in certain coordinates (x, y) could berelated to a two-dimensional spatial distribution of Stokes parameter inother coordinates (in the vicinity thereof).

5.2 Inverse Transformation by Generalized Inverse Matrix

As revealed from the above description, Expression (5.6) expresses alinear simultaneous equation, since column vector P on the left side isdetermined by measurement of the intensity distribution whereas thematrix R on the right side is fixed prior to measurement. Solving thislinear simultaneous equation leads to determination of column vector Q(unknown) on the right side. However, the number of elements of Q istypically considerably large as compared to the number of elements of P.(In the above example, the elements of Q are seven times larger innumber than the elements of P.) Hence, the matrix R does not have aninverse matrix.

As a method for solving a linear simultaneous equation written in matrixform in such a case, a method for using a generalized inverse matrix maybe employed. A matrix X that satisfies the following four conditions isreferred to as a generalized inverse matrix of R and denoted as R⁺.RXR=R   (5.9a)XRX=X   (5.9b)(RX)*=RX   (5.9c)(XR)*=XR   (5.9d)However, a superscript asterisk * added to the matrix denotes aconjugate transpose matrix. It should be noted that X as shown abovecertainly exists with respect to any R, and is further determineduniquely to R. In addition, a variety of methods have been proposed asconcrete methods for calculating R⁺ from R. (Reference: “Matrixnumerical value calculation”, written by Hayato Togawa, Ohmsha, Ltd.,1971, p46)

The use of this generalized inverse matrix R⁺ allows determination ofeach unknown element of column vector Q included in the right side ofExpression (5. 6), according to the following expression.Q=R ⁺ P   (5.10)That is, this means that the group of parameters p₀(x_(m), y_(n)),p_(c)(x_(m), y_(n)), p_(s)(x_(m)y_(n)), q_(ss)(x_(m), y_(n)),q_(c)(x_(m), y_(n)), q_(sc)(x_(m), y_(n)), and q_(cs)(x_(m), y_(n))(where m=1 . . . M, n=1 . . . N), which includes the two-dimensionalspatial distribution of Stokes parameters and the reference phasefunction differences, is obtained.

It is to be noted that, even in the case of using another matrixrepresentation as described at the end of the previous section, the useof an appropriate generalized inverse matrix enables fixing of “a listof information including two-dimensional spatial distribution of Stokesparameters of the light under measurement and phase attribute functionsof the measurement system.”

Each element obtained by this generalized inverse matrix is not inone-to-one correspondence with each quasi-sinusoidal vibration componentcontained in the intensity distribution obtained from the imagingelement. For example, as apparent from the above-mentioned derivationprocess, each of q_(ss)(x, y), q_(cc)(x, y), q_(sc)(x, y), and q_(cs)(x,y_(I)) is relative to both of two quasi-sinusoidal components related toφ₂(x, y)−φ₁(x, y) and φ₂(x, y)+φ₁(x, y).

That is to say, separation of elements by this generalized inversematrix calculation is not in one-to-one correspondence with separationof quasi-sinusoidal period components by frequency filtering which ismade by the Fourier transform method or the like.

5.3 Demodulation of Phase Attribute Function

Next, a phase attribute function is obtained from an element of columnvector Q.

As described as the general idea in the previous chapter, the phaseattribute function can be obtained as follows.

−φ₂(x, y) (or a function determined based thereupon) is obtained frominformation included in the intensity distribution obtained from theimaging element regardless of the SOP of the light under measurement.

−φ₁(x, y) as well as φ₂(x, y), and further a function relative to both,are obtained by the use of the relation between the φ₁(x, y) and theφ₂(x, y) (information provided in advance) regardless of the SOP of thelight under measurement.

By the use of an element of column vector Q obtained using a generalizedinverse matrix, an equation is further set up and solved so thatφ₁(x,y), φ₂(x, y) and functions equivalent to those, namely phaseattribute functions, can be obtained. Moreover, solving the results in asimultaneous manner, it is possible to determine the SOP of the lightunder measurement.

Concrete examples of a calculating expression in a case where eachelement of column vector Q is given by Expressions (5.7b) to (5.7h) areshown below. Those are only representation of results, but arecorresponded, when possible, to the methods described in Chapter 3.

A. Method for Obtaining Δφ₂(x, y), effective in case of S₁(x, y)≠0

Among the elements of Column Vector Q, p_(c)(x, y) and p_(s)(x, y) areobtained as follows. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 47} \right\rbrack & \quad \\{{I\left( {x,y} \right)} = {{p_{0}\left( {x,y} \right)} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{p_{c}\left( {x,y} \right)}} + {{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{p_{s}\left( {x,y} \right)}\underset{\_}{{{+ {\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}}{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{ss}\left( {x,y} \right)}} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{cc}\left( {x,y} \right)}}}} + \underset{\_}{{{+ {\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}}{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{sc}\left( {x,y} \right)}} + {{\cos\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{\sin\left\lbrack {\phi_{2}^{(i)}\left( {x,y} \right)} \right\rbrack}{q_{cs}\left( {x,y} \right)}}}}} & (5.3)\end{matrix}$From the above, Δφ₂(x, y) can be calculated as follows. $\begin{matrix}{{{\Delta\phi}_{2}\left( {x,y} \right)} = {{- \tan^{- 1}}\frac{p_{s}\left( {x,y} \right)}{p_{c}\left( {x,y} \right)}}} & (5.12)\end{matrix}$Here, the denominator and the numerator of the arctangent in the aboveexpression are both proportional to S₁(x, y) of the light undermeasurement. It is thus possible to obtain Δφ₂(x, y) according to theabove expression so long as S₁(x, y) is not zero.B. Method for Obtaining Δφ₂(x, y), Effective in Case where Either S₂(x,y)≠0 or S₃(x, y)≠0 is Satisfied

Among the elements of column vector Q, the following relations arederived even from those not used in above A. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 48} \right\rbrack & \quad \\{{\frac{1}{4}{m_{-}\left( {x,y} \right)}{{m_{+}\left( {x,y} \right)}\left\lbrack {{S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}} \right\rbrack}{\cos\left\lbrack {2{{\Delta\phi}_{2}\left( {x,y} \right)}} \right\rbrack}} = {{q_{ss}^{2}\left( {x,y} \right)} - {q_{cc}^{2}\left( {x,y} \right)} + {q_{sc}^{2}\left( {x,y} \right)} - {q_{cs}^{2}\left( {x,y} \right)}}} & \left( {5.13a} \right) \\{{{{\frac{1}{4}{m_{-}\left( {x,y} \right)}{{m_{+}\left( {x,y} \right)}\left\lbrack {{S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}} \right\rbrack}{\sin\left\lbrack {2{{\Delta\phi}_{2}\left( {x,y} \right)}} \right\rbrack}} = {2\left\lbrack {{{q_{cc}\left( {x,y} \right)}{q_{sc}\left( {x,y} \right)}} + {{q_{ss}\left( {x,y} \right)}{q_{cs}\left( {x,y} \right)}}} \right\rbrack}}{{{From}\quad{the}\quad{above}},\quad{{{\Delta\phi}_{2}\left( {x,y} \right)}\quad{can}\quad{be}\quad{calculated}\quad{as}\quad{{follows}.}}}}\quad} & \left( {5.13b} \right) \\{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}\tan^{- 1}\quad\frac{2\left\lbrack {{{q_{cc}\left( {x,y} \right)}{q_{sc}\left( {x,y} \right)}} + {{q_{ss}\left( {x,y} \right)}{q_{cs}\left( {x,y} \right)}}} \right\rbrack}{{q_{ss}^{2}\left( {x,y} \right)} - {q_{cc}^{2}\left( {x,y} \right)} + {q_{sc}^{2}\left( {x,y} \right)} - {q_{cs}^{2}\left( {x,y} \right)}}}} & (5.14)\end{matrix}$The denominator and the numerator of the arctangent in the aboveexpression are both proportional to S₂ ²(x, y)+S₃ ²(x, y) of the lightunder measurement. It is thus possible to obtain Δφ₂(x, y) according tothe above expression so long as S₂(x, y) and S₃(x, y) do notconcurrently become zero.C. Combination of A and B

Similar to the case (case of using the frequency filtering) described inChapter 3, adaptive combination of methods A and B is effective. Itshould be noted that, since a process to be performed for thecombination is completely the same as previously done, a descriptionthereof is omitted.

D. Method for Obtaining Δφ₂(x, y), Effective so Long as S₁, S₂, S₃ DoNot Concurrently Become Zero

Among the elements of column vector Q, the following expressions arederived from a further different combination. $\begin{matrix}\left\lbrack {{Mathematical}\quad{Expression}\quad 49} \right\rbrack & \quad \\{{\frac{1}{4}\left\{ {{{m_{2}^{2}\left( {x,y} \right)}\left\lbrack {S_{1}^{2}\left( {x,y} \right)} \right\rbrack} + {m_{-}\left( {x,y} \right)} + {{m_{-}\left( {x,y} \right)}{{m_{+}\left( {x,y} \right)}\left\lbrack {{S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}} \right\rbrack}}} \right\}{\cos\left\lbrack {2\Delta\quad{\phi_{2}\left( {x,y} \right)}} \right\rbrack}} = {{p_{c}^{2}\left( {x,y} \right)} - {p_{s}^{2}\left( {x,y} \right)} + {q_{ss}^{2}\left( {x,y} \right)} - {q_{cc}^{2}\left( {x,y} \right)} + {q_{sc}^{2}\left( {x,y} \right)} - {q_{cs}^{2}\left( {x,y} \right)}}} & \left( {5.15a} \right) \\{{\frac{1}{4}\left\{ {{{m_{2}^{2}\left( {x,y} \right)}\left\lbrack {S_{1}^{2}\left( {x,y} \right)} \right\rbrack} + {m_{-}\left( {x,y} \right)} + {{m_{-}\left( {x,y} \right)}{{m_{+}\left( {x,y} \right)}\left\lbrack {{S_{2}^{2}\left( {x,y} \right)} + {S_{3}^{2}\left( {x,y} \right)}} \right\rbrack}}} \right\}{\sin\left\lbrack {2\Delta\quad{\phi_{2}\left( {x,y} \right)}} \right\rbrack}} = {2\left\lbrack {{{- {p_{c}\left( {x,y} \right)}}{p_{s}\left( {x,y} \right)}} + {{q_{cc}\left( {x,y} \right)}{q_{sc}\left( {x,y} \right)}} + {{q_{ss}\left( {x,y} \right)}{q_{cs}\left( {x,y} \right)}}} \right\rbrack}} & \left( {5.15b} \right)\end{matrix}$From the above, an expression is derived as follows as a thirdexpression for calculating the Δφ₂(x, y). $\begin{matrix}{{{\Delta\phi}_{2}\left( {x,y} \right)} = {\frac{1}{2}\tan^{- 1}\frac{\begin{matrix}{2\left\lbrack {{{- {p_{c}\left( {x,y} \right)}}p_{s}\left( {x,y} \right)} + {{q_{cc}\left( {x,y} \right)}{q_{sc}\left( {x,y} \right)}} +} \right.} \\\left. {{q_{ss}\left( {x,y} \right)}{q_{cs}\left( {x,y} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{{p_{c}^{2}\left( {x,y} \right)} - {p_{s}^{2}\left( {x,y} \right)} + {q_{ss}^{2}\left( {x,y} \right)} - {q_{cc}^{2}\left( {x,y} \right)} +} \\{{q_{sc}^{2}\left( {x,y} \right)} - {q_{cs}^{2}\left( {x,y} \right)}}\end{matrix}}}} & (5.16)\end{matrix}$The denominator and the numerator of the arctangent in the aboveexpression are both proportional to m₂ ²(x, y)S₁ ²(x, y)+m⁻(x, y)m₊(x,y)[S₂ ²(x, y)+S₃ ²(x, y)]. It is thus possible to obtain Δφ₂(x, y)according to the above expression so long as S₁(x, y), S₂(x, y) andS₃(x, y) do not concurrently become zero.

It should be noted that “S₁(x, y)=S₂(x, y)=S₃(x, y)=0” is satisfied whenthe light under measurement is unpolarized light. In this case,calibration of the phase attribute function itself is not requiredbecause only the polarization degree (i.e. 0) is significantinformation.

E. Calculation of Δφ₁(x, y)

Since fluctuations in Δφ₁(x, y) are considered to be similar to those inΔφ₂(x, y), it is possible to obtain Δφ₁(x, y) by proportionalcalculation (e.g. by using a relation between Δφ₂(x, y) and Δφ₁(x, y))from a measured value of Δφ₂(x, y).

F. Demodulation of Two-Dimensional Spatial Distribution of StokesParameters

Using obtained Δφ₁(x, y) and Δφ₂(x, y), the two-dimensional spatialdistributions S₀(x, y), S₁(x, y), S₂(x, y), and S₃(x, y) of Stokesparameters are determined from p₀(x, y), p_(c)(x, y), p_(s)(x, y),q_(ss)(x, y), q_(cc)(x, y), qsc(x, y), and q_(cs)(x, y). For example,the following expressions may be used. $\begin{matrix}{\left\lbrack {{Mathematical}{\quad\quad}{Expression}\quad 50} \right\rbrack{{S_{0}\left( {x,y} \right)} = {\frac{2}{m_{0}\left( {x,y} \right)}{p_{0}\left( {x,y} \right)}}}} & \left( {5.17a} \right) \\{{S_{1}\left( {x,y} \right)} = {\frac{2}{m_{2}\left( {x,y} \right)}\left\lbrack {{{p_{c}\left( {x,y} \right)}\cos\quad\Delta\quad{\phi_{2}\left( {x,y} \right)}} - {{p_{s}\left( {x,y} \right)}\sin\quad\Delta\quad{\phi_{2}\left( {x,y} \right)}}} \right\rbrack}} & \left( {5.17b} \right) \\{{S_{2}\left( {x,y} \right)} = {\frac{2}{{m\_}\left( {x,y} \right)}\left\{ {{\left\lbrack {{q_{ss}\left( {x,y} \right)} + {q_{cc}\left( {x,y} \right)}} \right\rbrack{\cos\left\lbrack {{\Delta\quad{\phi_{2}\left( {x,y} \right)}} - {\Delta\quad{\phi_{1}\left( {x,y} \right)}}} \right\rbrack}} - \left. \quad{\left\lbrack {{q_{sc}\left( {x,y} \right)} - {q_{cs}\left( {x,y} \right)}} \right\rbrack{\sin\left\lbrack {{\Delta\quad{\phi_{2}\left( {x,y} \right)}} - {\Delta\quad{\phi_{1}\left( {x,y} \right)}}} \right\rbrack}} \right\}} \right.}} & \left( {5.17c} \right) \\{{S_{3}\left( {x,y} \right)} = {\frac{2}{{m\_}\left( {x,y} \right)}\left\{ {{{- \left\lbrack {{q_{ss}\left( {x,y} \right)} + {q_{cc}\left( {x,y} \right)}} \right\rbrack}{\sin\left\lbrack {{\Delta\quad{\phi_{2}\left( {x,y} \right)}} - {\Delta\quad{\phi_{1}\left( {x,y} \right)}}} \right\rbrack}} - \left. \quad{\left\lbrack {{q_{sc}\left( {x,y} \right)} - {q_{cs}\left( {x,y} \right)}} \right\rbrack{\cos\left\lbrack {{\Delta\quad{\phi_{2}\left( {x,y} \right)}} - {\Delta\quad{\phi_{1}\left( {x,y} \right)}}} \right\rbrack}} \right\}} \right.}} & \left( {5.17d} \right)\end{matrix}$

EXAMPLE 1

In the following, a preferred example of the present invention isspecifically described with reference to FIGS. 12 to 18. FIG. 12 shows aconstitutional view of one example of an imaging polarimeter. As shownin this figure, this device comprises a photo-projecting side unit 200and a photo-receiving side unit 300. It is to be noted that numeral 400denotes a sample.

The photo-projecting side unit 200 comprises: a power source 201; alight source 202 that is turned on by power feeding from the powersource 201; a pinhole plate 203 arranged on the front face side of thelight source 202 in the light-projecting direction; a collimator lens204 for collimating light transmitting through the pinhole of thepinhole plate 203; a shutter 205 which is arranged on the front faceside of the collimator lens 204 and opens and closes to transmit orblock the transmitted light; and a polarizer 206 on which the lighthaving transmitted through the shutter is incident.

The light after passage of a polarizer 206 is projected from thephoto-projecting side unit 200 onto the sample 400. The lighttransmitted through or reflected on the sample 400 is incident on thephoto-receiving side unit 300.

On an incident light channel in the photo-receiving side unit 300, animaging lens 301 and a polarimetric imaging device 310 are arranged.This polarimetric imaging device 310 is so constituted that twobirefringent prism pairs 302 a and 302 b, an analyzer 303 and a CCDimaging element 304 intervene in sequence. The photo-receiving side unit300 further comprises an AND converter 305 that converts a light outputfrom the CCD imaging element 304 to a digital signal. The digital outputsignal from the A/D converter 305 is processed in a computer 306 such asa personal computer (PC).

As widely known, the computer 306 comprises: an arithmetic processingpart 306 a comprised of a microprocessor and the like; a memory part 306b comprised of an ROM, a RAM, an HDD and the like; and a measurementresult output part 306 c comprised of a display, a printer, a variety ofdata output devices, a communication device, and the like.

In addition, although the photo-projecting side unit and thephoto-receiving side unit are separately constituted in this example asdescribed above, the photo-projecting side unit and the photo-receivingunit may be integrated. Furthermore, the light source 202 emitssingle-wavelength light and includes a laser, a white lamp+aninterference filter, a lamp having bright line spectrum+a color filter,and the like.

FIG. 13 is a sectional view showing the polarimetric imaging device 310shown in FIG. 12. As shown in this figure, a glass plate 307 is arrangedon the front side of the birefringent prism pair 302 a and a spacer 308and a cover glass 309 are arranged between the analyzer 303 and the CCDimaging element 304. In this constitution, a completely compactpolarimetric imaging device integrated with the imaging element can beimplemented.

Next, FIG. 14 shows a flowchart of a pre-calibration process. As shownin this figure, as the pre-calibration process, first in Step 1401,light whose two-dimensional spatial distribution of Stokes parameters isknown is incident on a device (photo-receiving side unit 300 in thiscase). It should be noted that, for generation of light whosetwo-dimensional spatial distribution of Stokes parameters is known, forexample, the polarizer 206 of the device in the figure may be rotated soas to be arranged in a desired orientation.

Next in Step 1402, a two-dimensional spatial intensity distribution oftransmitted light is measured with the imaging element. Here, theshutter 205 may be utilized for reduction in influence of unnecessarylight, such as lost light. Specifically, an intensity distribution ofthe unnecessary light can be canceled out by taking a difference inintensity distribution when measured with the shutter open and whenmeasured with the shutter closed.

Next in Step 1403, the two-dimensional spatial intensity distribution ofthe transmitted light is forwarded from the imaging element to thecomputer 306, to be provided to processing in the arithmetic processingpart 306 a.

Next in Step 1404, reference phase functions and reference amplitudefunctions are calculated by an action of the arithmetic processing part306 a.

Next in Step 1405, the calculated reference phase functions andreference amplitude functions are stored into the memory part 306 b,whereby the pre-calibration process is completed.

FIG. 15 shows a flowchart of a measurement process. As shown in thefigure, as the measurement process, first, light under measurement isincident on the device in Step 1501. Here, when the aim of measurementis to examine the SOP associated with transmission and reflection of thelight through and on the sample 400, first, the sample 400 is irradiatedwith light whose SOP is known, and then the light transmitted through orreflected on the sample 400 is incident on the device (photo-receivingside unit 300: polarimeter).

Next in Step 1502, the two-dimensional spatial intensity distribution ofthe transmitted light is measured with the imaging element 304. Here,the shutter 205 can be utilized for reduction in influence ofunnecessary light, such as lost light. Specifically, the intensitydistribution of the unnecessary light can be canceled out by taking adifference in intensity distribution when measured with the shutter openand when measured with the shutter closed.

Next in Step 1503, the two-dimensional spatial intensity distribution ofthe transmitted light is forwarded from the imaging element 304 to thecomputer 306, to be provided to processing in the arithmetic processingpart 306 a.

Next in Step 1504, in the computer 306, the arithmetic processing part306 a acquires reference phase functions and reference amplitudefunctions from the memory part 306 b.

Next in Step 1505, in the computer 306, the arithmetic processing part306a calculates reference phase function differences (Δφ₁ and Δφ₂) bythe use of the measured two-dimensional spatial intensity distribution,the reference phase functions and the reference amplitude functions.

Next in Step 1506, in the computer 306, the arithmetic processing part306 a calculates two-dimensional spatial distribution of Stokesparameters of the light under measurement by the use of the measuredtwo-dimensional spatial intensity distribution, and differences of thereference phase functions and the reference amplitude functions.

Next in Step 1507, in the computer 306, the arithmetic processing part306 a outputs the two-dimensional spatial distribution of the Stokesparameters of the light under measurement. Examples of the measurementresult output part 306 c may include a memory, a hard disc, and otherprocessing part (calculating part for ellipticity angle, azimuth angle,etc.).

As described above, in the imaging polarimeter of this example, theStokes parameters regarding the light under measurement are calculatedthrough the pre-calibration process shown in FIG. 14 and the measurementprocess shown in FIG. 15 in the system constitution shown in FIG. 12.

An example of specific experimental results is described with referenceto FIGS. 16 to 18. In this experiment, pre-calibration of the referencephase function and the like is performed. Then, measurement is taken 3times, rising the temperature about 3° C. at a time. The light undermeasurement is a linear polarized light with an azimuth of 22.5° and itsSOP is constant in a measurement region.

FIGS. 16 and 17 show graphs showing measured results of the ellipticityangle. In FIG. 16, the measured result in the case of thepre-calibration only is shown. FIG. 6A shows the measured result in casethat the temperature is raised 3° C., FIG. 6B shows the measured resultin case that the temperature is further raised 3° C. (+6° C.), and FIG.6C shows the measured result in case that the temperature is furtherraised 3° C. (+9° C.). In FIG. 17, the measured result in case thatcalibration is performed during measurement in addition to thepre-calibration, and similar to FIG. 16, the measured results in casethat the temperature is raised 3° C., 6° C. and 9° C. are shown in FIGS.17A, 17B and 17C, respectively.

As shown in FIG. 16, it is found that the measured values are shifted asthe temperature is raised in the case of the pre-calibration only.Although the SOP of the light under measurement is constant, themeasured values are shifted together with the temperature change. Inaddition, this shift amount varies depending upon a position (spatialcoordinates).

Meanwhile, as shown in FIG. 17, in case that the difference of thereference phase function is corrected during measurement (correctionduring measurement of Δφ₁ and Δφ₂) in addition to the pre-calibration,almost an ideal value (0°) is provided over the whole measurement regionwithout depending on the temperature change.

FIG. 18A shows a sectional view of the graph showing the measured resultin FIG. 16 in case that only the pre-calibration is performed, and FIG.18B shows a sectional view of the graph showing the measured result inFIG. 17 in case that the calibration during measurement is performed inaddition to the pre-calibration. As can be seen from FIG. 18, when thecalibration (correction) during measurement is added to thepre-calibration, the measured value does not depend on the temperaturechange.

Here, since the light under measurement is a linearly polarized lightwith an azimuth of 22.5°, the measured value of the ellipticity angleshould be 0 ideally without depending on the position (spatialcoordinates). As can be seen from FIGS. 16 to 18, while the measuredresult is varied as the temperature is raised in the case of thepre-calibration only, the measured result is in the vicinity of 0° evenwhen the temperature is raised in case of the calibration duringmeasurement is performed in addition to the pre-calibration. Thus, inthe imaging polarimeter, the measured result can be stably providedwithout depending upon a temperature change.

1. An imaging polarimetry, comprising: a step of preparing apolarimetric imaging device, where a first birefringent prism pair, asecond birefringent prism pair and an analyzer, through which lightunder measurement passes in sequence, and a device for obtaining atwo-dimensional intensity distribution of the light having passedthrough the analyzer are provided, each birefringent prism paircomprises parallel flat plates in which two wedge-shaped retardershaving the same apex angle are attached and directions of fast axes ofthe two retarders are orthogonal to each other, the second birefringentprism pair is arranged such that the direction of a principal axis ofthe second birefringent prism pair disagrees with the direction of aprincipal axis of the first birefringent prism pair, and the analyzer isarranged such that the direction of a transmission axis of the analyzerdisagrees with the direction of the principal axis of the secondbirefringent prism pair; a step of launching the light under measurementinto the polarimetric imaging device to obtain a two-dimensionalintensity distribution; and an arithmetic step of obtaining a set ofphase attribute functions of a measurement system, and also obtaining aparameter indicating a two-dimensional spatial distribution of a stateof polarization (SOP) of the light under measurement by the use of theabove obtained intensity distribution, wherein the set of phaseattribute functions is a set of functions defined by properties of thepolarimetric imaging device, and includes a function depending upon atleast a first reference phase function (φ₁(x, y)) as retardation of thefirst birefringent prism pair and a function depending upon at least asecond reference phase function (φ₂(x, y)) as retardation of the secondbirefringent prism pair, and by those functions themselves, or byaddition of another function defined by the properties of thepolarimetric imaging device, the set of phase attribute functionsbecomes a set of functions sufficient to determine the parameterindicating the two-dimensional spatial distribution of the SOP of thelight under measurement.
 2. The imaging polarimetry according to claim1, wherein the analyzer is arranged such that the direction of thetransmission axis of the analyzer forms an angle of 45° with respect tothe direction of the principal axis of the second birefringent prismpair.
 3. The imaging polarimetry according to claim 1, wherein, in thearithmetic step, the set of phase attribute functions is composed of thefirst reference phase function and the second reference phase function,and data showing a relation between the first reference phase functionand the second reference phase function is made available, and thearithmetic step is a unit where, by the use of the obtained intensitydistribution, a first intensity distribution component whichnonperiodically vibrates with special coordinates and a third intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon the second reference phase function and notdepending upon the first reference phase function are obtained, and atleast one of a second intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon a differencebetween the first reference phase function and the second referencephase function, a fourth intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon a sum of thefirst reference phase function and the second reference phase function,and a fifth intensity distribution component which vibrates with spatialcoordinates at a frequency depending upon the first reference phasefunction and not depending upon the second reference phase function isobtained, and by the use of the data showing the relation between thefirst reference phase function and the second reference phase functionand each of the intensity distribution components, the first referencephase function and the second reference phase function are obtained, andalso the parameter indicating the two-dimensional spatial distributionof the SOP is obtained.
 4. The imaging polarimetry according to claim 1,wherein, in the arithmetic step, the set of phase attribute functions iscomposed of a difference (Δφ₁(x, y)) of the first reference phasefunction from a reference value for calibration of the first referencephase function and a difference (Δφ₂(x, y)) of the second referencephase function from a reference value for calibration of the secondreference phase function, and the reference value (φ₁ ^((i))(x, y)) forcalibration of the first reference phase function, the reference value(φ₂ ^((i))(x, y)) for calibration of the second reference phasefunction, and data showing a relation between the first reference phasefunction difference and the second reference phase function differenceare made available, and the arithmetic step is a unit where, by the useof the obtained intensity distribution, a first spectral intensitycomponent which nonperiodically vibrates with spatial coordinates and athird intensity distribution component which vibrates with spatialcoordinates at a frequency depending upon the second reference phasefunction and not depending upon the first reference phase function areobtained, and at least one of a second intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon adifference between the first reference phase function and the secondreference phase function, a fourth intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon asum of the first reference phase function and the second reference phasefunction, and a fifth intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon the firstreference phase function and not depending upon the second referencephase function is obtained, and by the use of the reference value forcalibration of the first reference phase function, the reference valuefor calibration of the second reference phase function, the data showingthe relation between the first reference phase function difference andthe second reference phase function difference, and each of the obtainedintensity distribution components, the first reference phase functiondifference and the second reference phase function difference areobtained, and also the parameter indicating the two-dimensional spatialdistribution of the SOP is obtained.
 5. The imaging polarimetryaccording to claim 4, further comprising a step of launching light forcalibration, with known parameters each showing the two-dimensionalspatial distribution of the SOP, into the polarimetric imaging device toobtain a two-dimensional intensity distribution for calibration, so asto obtain the reference value (φ₁ ^((i))(x, y)) for calibration of thefirst reference phase function and the reference value (φ₂ ^((i))(x, y))for calibration for the second reference phase function by the use ofeach of the parameters showing the two-dimensional spatial distributionof the SOP of the light for calibration and the obtained intensitydistribution for calibration, whereby these reference values forcalibration are made available.
 6. The imaging polarimetry according toclaim 4, further comprising a step of launching light for calibration,with known parameters each showing the two-dimensional spatialdistribution of the SOP, into the polarimetric imaging device to obtaina two-dimensional intensity distribution for calibration, so as toobtain the reference value (φ₁ ^((i))(x, y)) for calibration of thefirst reference phase function, the reference value (φ² ^((i))(x, y))for calibration for the second reference phase function, and the datashowing the relation between the first reference phase functiondifference and the second reference phase function difference, by theuse of each of the parameters showing the two-dimensional spatialdistribution of the SOP of the light for calibration and the obtainedintensity distribution for calibration, whereby these reference valuesfor calibration are made available.
 7. The imaging polarimetry accordingto claim 3, further comprising a step of launching light forcalibration, with known parameters each showing the two-dimensionalspatial distribution of the SOP, into the polarimetric imaging device toobtain a two-dimensional intensity distribution for calibration, so asto obtain the data showing the relation between the first referencephase function difference and the second reference phase functiondifference, by the use of each of the parameters showing thetwo-dimensional spatial distribution of the SOP of the light forcalibration and the obtained intensity distribution for calibration,whereby the data showing the relation between the first reference phasefunction difference and the second reference phase function differenceis made available.
 8. The imaging polarimetry according to claim 5,wherein the light for calibration is linearly polarized light.
 9. Theimaging polarimetry according to claim 7, wherein the light forcalibration is linearly polarized light.
 10. The imaging polarimetryaccording to claim 1, wherein, in the arithmetic step, a value of eachelement of a generalized inverse matrix of a matrix is made availablesuch that a relation is formed where a first vector includinginformation on the two-dimensional intensity distribution is expressedby a product of the matrix and a second vector including information onthe two-dimensional spatial distribution of the SOP of the light undermeasurement and information on the set of phase attribute functions, andthe arithmetic step is a unit where a value of each element of the firstvector is specified by the use of the obtained intensity distribution, avalue of each element of the second vector is obtained by calculation ofa product of the generalized inverse matrix and the first vector, and bythe use of the value of the element included in the second vector, theset of phase attribute functions is obtained, and also the parametershowing the two-dimensional spatial distribution of the SOP of the lightunder measurement is obtained.
 11. The imaging polarimetry according toclaim 10, wherein, in the arithmetic step, the set of phase attributefunctions is composed of a difference (Δφ₁(x, y)) of the first referencephase function from a reference value for calibration of the firstreference phase function and a difference (Δφ₂(x, y)) of the secondreference phase function from a reference value for calibration of thesecond reference phase function, data showing a relation between thefirst reference phase function difference and the second reference phasefunction difference is made available, and further the generalizedinverse matrix of the matrix, obtained from the reference value (φ₁^((i))(x, y)) for calibration of the first reference phase function andthe reference value (φ₂ ^((i))(x, y)) for calibration for the secondreference phase function, is made available, and the arithmetic step isa unit where a value of each element of the first vector is specified bythe use of the obtained intensity distribution, a value of each elementof the second vector is obtained by calculation of a product of thegeneralized inverse matrix and the first vector, and by the use of thevalue of the element included in the second vector and the data showingthe relation between the first reference phase function difference andthe second reference phase function difference, the first referencephase function difference and the second reference phase functiondifference are obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP is obtained.
 12. Animaging polarimeter, comprising: a polarimetric imaging device in whicha first birefringent prism pair, a second birefringent prism pair and ananalyzer, through which light under measurement passes in sequence, anda device for obtaining a two-dimensional intensity distribution of thelight having passed through the analyzer are provided, each birefringentprism pair comprises parallel flat plates in which two wedge-shapedretarders having the same apex angle are attached and directions of fastaxes of the two retarders are orthogonal to each other, the secondbirefringent prism pair is arranged such that the direction of aprincipal axis of the second birefringent prism pair disagrees with thedirection of a principal axis of the first birefringent prism pair, andthe analyzer is arranged such that the direction of a transmission axisof the analyzer disagrees with the direction of the principal axis ofthe second birefringent prism pair; and an arithmetic unit for obtaininga set of phase attribute functions of a measurement system, and alsoobtaining a parameter indicating a two-dimensional spatial distributionof a state of polarization (SOP) of the light under measurement by theuse of the two-dimensional intensity distribution obtained by launchingthe light under measurement into the polarimetric imaging device,wherein the set of phase attribute functions is a set of functionsdefined by properties of the polarimetric imaging device, and includes afunction depending upon at least a first reference phase function (φ₁(x,y)) as retardation of the first birefringent prism pair and a functiondepending upon at least a second reference phase function (φ₂(x, y)) asretardation of the second retarder, and by those functions themselves,or by addition of another function defined by the properties of thepolarimetric imaging device, the set of phase attribute functionsbecomes a set of functions sufficient to determine a parameterindicating a two-dimensional spatial distribution of the SOP of thelight under measurement.
 13. The imaging polarimeter according to claim12, wherein the analyzer is arranged such that the direction of thetransmission axis of the analyzer forms an angle of 45° with respect tothe direction of the principal axis of the second birefringent prismpair.
 14. The spectroscopic polarimeter according to claim 12, wherein,in the arithmetic unit, the set of phase attribute functions is composedof the first reference phase function and the second reference phasefunction, and data showing a relation between the first reference phasefunction and the second reference phase function is made available, andthe arithmetic unit is a unit where, by the use of the two-dimensionalintensity distribution obtained by launching the light under measurementinto the polarimetric imaging device, a first intensity distributioncomponent which nonperiodically vibrates with spatial coordinates and athird intensity distribution component which vibrates with spatialcoordinates at a frequency depending upon a second reference phasefunction and not depending upon the first reference phase function areobtained, and at least one of a second intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon adifference between the first reference phase function and the secondreference phase function, a fourth intensity distribution componentwhich vibrates with spatial coordinates at a frequency depending upon asum of the first reference phase function and the second reference phasefunction, and a fifth intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon the firstreference phase function and not depending upon the second referencephase function is obtained, and by the use of the data showing therelation between the first reference phase function and the secondreference phase function and each of the obtained intensity distributioncomponents, the first reference phase function and the second referencephase function are obtained, and also the parameter indicating thetwo-dimensional spatial distribution of the SOP is obtained.
 15. Theimaging polarimeter according to claim 12, wherein, in the arithmeticunit, the set of phase attribute functions is composed of a difference(Δφ₁(x, y)) of the first reference phase function from a reference valuefor calibration of the first reference phase function and a difference(Δφ₂(x, y)) of the second reference phase function from a referencevalue for calibration of the second reference phase function, and thereference value (φ₁ ^((i))(x, y)) for calibration of the first referencephase function, the reference value (φ₂ ^((i))(x, y)) for calibration ofthe second reference phase function, and data showing a relation betweenthe first reference phase function difference and the second referencephase function difference are made available, and the arithmetic unit isa unit where, by the use of the two-dimensional intensity distributionobtained by launching the light under measurement into the polarimetricimaging device, a first intensity distribution component whichnonperiodically vibrates with spatial coordinates and a third intensitydistribution component which vibrates with spatial coordinates at afrequency depending upon a second reference phase function and notdepending upon the first reference phase function are obtained, and atleast one of a second intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon a differencebetween the first reference phase function and the second referencephase function, a fourth intensity distribution component which vibrateswith spatial coordinates at a frequency depending upon a sum of thefirst reference phase function and the second reference phase function,and a fifth intensity distribution component which vibrates with spatialcoordinates at a frequency depending upon the first reference phasefunction and not depending upon the second reference phase function isobtained, and by the use of the reference value for calibration of thefirst reference phase function, the reference value for calibration ofthe second reference phase function, the data showing the relationbetween the first reference phase function difference and the secondreference phase function difference, and each of the obtained intensitydistribution components, the first reference phase function differenceand the second reference phase function difference are obtained, andalso the parameter indicating the two-dimensional spatial distributionof the SOP is obtained.
 16. The imaging polarimeter according to claim12, wherein, in the arithmetic unit, a value of each element of ageneralized inverse matrix of a matrix is made available such that arelation is formed where a first vector including information on thetwo-dimensional intensity distribution is expressed by a product of thematrix and a second vector including information on the two-dimensionalspatial distribution of the SOP of the light under measurement andinformation on the set of the phase attribute function, and thearithmetic unit is a unit where a value of each element of the firstvector is specified by the use of the two-dimensional intensitydistribution obtained by launching the light under measurement into thepolarimetric imaging device, a value of each element of the secondvector is obtained by calculation of a product of the generalizedinverse matrix and the first vector, and by the use of the value of theelement included in the second vector, the set of phase attributefunctions is obtained, and also the parameter showing thetwo-dimensional spatial distribution of the SOP of the light undermeasurement is obtained.
 17. The spectroscopic polarimeter according toclaim 16, wherein, in the arithmetic unit, the set of phase attributefunctions is composed of a difference (Δφ₁(x, y)) of the first referencephase function from a reference value for calibration of the firstreference phase function and a difference (Δφ₂(x, y)) of the secondreference phase function from a reference value for calibration of thesecond reference phase function, data showing a relation between thefirst reference phase function difference and the second reference phasefunction difference is made available, and further the generalizedinverse matrix of the matrix, obtained from the reference value (φ₁^((i))(x, y)) for calibration of the first reference phase function andthe reference value (φ₂ ^((i))(x, y)) for calibration for the secondreference phase function, is made available, and the arithmetic unit isa unit where a value of each element of the first vector is specified bythe use of the two-dimensional intensity distribution obtained bylaunching the light under measurement into the polarimetric imagingdevice, a value of each element of the second vector is obtained bycalculation of a product of the generalized inverse matrix and the firstvector, and by the use of the value of the element included in thesecond vector and the data showing the relation between the firstreference phase function difference and the second reference phasefunction difference, the first reference phase function difference andthe second reference phase function difference are obtained, and alsothe parameter showing the two-dimensional spatial distribution of theSOP is obtained.